| L(s) = 1 | + 14·5-s − 17·9-s − 32·11-s + 28·13-s + 154·17-s − 224·19-s − 68·23-s + 45·25-s − 236·29-s − 196·31-s + 346·37-s + 420·41-s + 344·43-s − 238·45-s + 84·47-s − 438·53-s − 448·55-s − 56·59-s − 98·61-s + 392·65-s − 336·67-s − 896·71-s − 966·73-s + 52·79-s − 440·81-s + 392·83-s + 2.15e3·85-s + ⋯ |
| L(s) = 1 | + 1.25·5-s − 0.629·9-s − 0.877·11-s + 0.597·13-s + 2.19·17-s − 2.70·19-s − 0.616·23-s + 9/25·25-s − 1.51·29-s − 1.13·31-s + 1.53·37-s + 1.59·41-s + 1.21·43-s − 0.788·45-s + 0.260·47-s − 1.13·53-s − 1.09·55-s − 0.123·59-s − 0.205·61-s + 0.748·65-s − 0.612·67-s − 1.49·71-s − 1.54·73-s + 0.0740·79-s − 0.603·81-s + 0.518·83-s + 2.75·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.994892827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.994892827\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 17 T^{2} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 14 T + 151 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 32 T + 1105 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 28 T + 3998 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 154 T + 15163 T^{2} - 154 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 224 T + 25929 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 68 T + 23677 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 236 T + 33694 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 196 T + 67373 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 346 T + 65967 T^{2} - 346 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 420 T + 176614 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 84 T + 201085 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 438 T + 280447 T^{2} + 438 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 56 T + 360889 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 98 T + 253751 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 336 T + 627937 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 896 T + 800494 T^{2} + 896 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 966 T + 187 p^{2} T^{2} + 966 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 52 T + 332261 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 392 T + 895462 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 294 T + 1298347 T^{2} + 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 420 T + 1655734 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.27964200336954694958556678218, −9.703539498562822683337976313806, −9.173095844843444077270337112791, −9.035641625722344585351170423949, −8.398096148381182884498805131543, −8.004761630587589296442409993244, −7.41952979003001832770637516285, −7.40153099518458197574105199548, −6.26150272445885971255217633353, −6.00723360145756718275011750517, −5.71733335549285427546974616114, −5.66710954099127996557867567341, −4.59537102160107639425333597526, −4.34818989376491098157924747890, −3.54835392666532120035136860562, −3.08026440410784046737423325332, −2.27172424655738560167013465477, −2.06859022429849064430480748692, −1.28965645719847951117285017960, −0.37471338841160090817318730000,
0.37471338841160090817318730000, 1.28965645719847951117285017960, 2.06859022429849064430480748692, 2.27172424655738560167013465477, 3.08026440410784046737423325332, 3.54835392666532120035136860562, 4.34818989376491098157924747890, 4.59537102160107639425333597526, 5.66710954099127996557867567341, 5.71733335549285427546974616114, 6.00723360145756718275011750517, 6.26150272445885971255217633353, 7.40153099518458197574105199548, 7.41952979003001832770637516285, 8.004761630587589296442409993244, 8.398096148381182884498805131543, 9.035641625722344585351170423949, 9.173095844843444077270337112791, 9.703539498562822683337976313806, 10.27964200336954694958556678218
