| L(s) = 1 | − 31·2-s + 239·4-s − 1.25e3·5-s + 1.41e4·7-s − 899·8-s + 3.87e4·10-s + 2.15e4·11-s + 2.42e4·13-s − 4.37e5·14-s + 8.51e4·16-s + 1.56e5·17-s − 9.58e4·19-s − 2.98e5·20-s − 6.66e5·22-s + 7.35e5·23-s + 1.17e6·25-s − 7.52e5·26-s + 3.37e6·28-s + 2.67e6·29-s + 1.07e7·31-s + 2.34e6·32-s − 4.86e6·34-s − 1.76e7·35-s + 2.19e7·37-s + 2.97e6·38-s + 1.12e6·40-s − 2.60e7·41-s + ⋯ |
| L(s) = 1 | − 1.37·2-s + 0.466·4-s − 0.894·5-s + 2.22·7-s − 0.0775·8-s + 1.22·10-s + 0.443·11-s + 0.235·13-s − 3.04·14-s + 0.325·16-s + 0.455·17-s − 0.168·19-s − 0.417·20-s − 0.606·22-s + 0.547·23-s + 3/5·25-s − 0.323·26-s + 1.03·28-s + 0.703·29-s + 2.09·31-s + 0.394·32-s − 0.624·34-s − 1.98·35-s + 1.92·37-s + 0.231·38-s + 0.0694·40-s − 1.44·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.614355312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.614355312\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 31 T + 361 p T^{2} + 31 p^{9} T^{3} + p^{18} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 288 p^{2} T + 2107886 p^{2} T^{2} - 288 p^{11} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 21512 T + 1790961254 T^{2} - 21512 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1868 p T + 5870669214 T^{2} - 1868 p^{10} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 156956 T + 212670095078 T^{2} - 156956 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 95896 T + 629883192438 T^{2} + 95896 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 31968 p T + 3363187908526 T^{2} - 31968 p^{10} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2678212 T + 15397908029438 T^{2} - 2678212 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10782432 T + 69294691361342 T^{2} - 10782432 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 21968332 T + 359210373327534 T^{2} - 21968332 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 26060372 T + 693239183881142 T^{2} + 26060372 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7191160 T + 750634586008230 T^{2} + 7191160 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 671920 p T + 382042606129310 T^{2} - 671920 p^{10} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3131116 T + 6584849973489806 T^{2} + 3131116 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 35494664 T + 7388006896329158 T^{2} - 35494664 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 341497340 T + 52053805546777278 T^{2} - 341497340 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 288195816 T + 74605839041196758 T^{2} + 288195816 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 210286064 T + 91549406631588686 T^{2} + 210286064 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 232663084 T + 130536207391012086 T^{2} + 232663084 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 24755040 T + 43090694479668638 T^{2} + 24755040 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 372082152 T + 379908828789982198 T^{2} - 372082152 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 427639116 T + 702028302670151638 T^{2} - 427639116 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1771658884 T + 2207048700436243398 T^{2} - 1771658884 p^{9} T^{3} + p^{18} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.32602752364315288338021734041, −13.72342347554983072557557268868, −12.91364408409424783332083271215, −11.88946166246651796551249961640, −11.60856379249352492817263791990, −11.37204484292808353647540385221, −10.28302127046966178966837884250, −10.10646935639094541940108849076, −8.812280147862906492745604514838, −8.748233182407995189558827935049, −7.990582164272530402805973693704, −7.80400228166169182857021768609, −6.85214808409754329638845821226, −5.85371723475241735936610684749, −4.60354445673882847155388515795, −4.56785295714959469379177249519, −3.21509233838646124568201381732, −2.05330947350206793785877014736, −0.896226279663258046420960005613, −0.840875712404553090896598392552,
0.840875712404553090896598392552, 0.896226279663258046420960005613, 2.05330947350206793785877014736, 3.21509233838646124568201381732, 4.56785295714959469379177249519, 4.60354445673882847155388515795, 5.85371723475241735936610684749, 6.85214808409754329638845821226, 7.80400228166169182857021768609, 7.990582164272530402805973693704, 8.748233182407995189558827935049, 8.812280147862906492745604514838, 10.10646935639094541940108849076, 10.28302127046966178966837884250, 11.37204484292808353647540385221, 11.60856379249352492817263791990, 11.88946166246651796551249961640, 12.91364408409424783332083271215, 13.72342347554983072557557268868, 14.32602752364315288338021734041
