| L(s) = 1 | − 8·2-s + 48·4-s − 53·5-s − 256·8-s + 424·10-s − 191·11-s + 379·13-s + 1.28e3·16-s − 340·17-s + 1.76e3·19-s − 2.54e3·20-s + 1.52e3·22-s − 3.23e3·23-s − 1.74e3·25-s − 3.03e3·26-s − 4.45e3·29-s − 1.99e3·31-s − 6.14e3·32-s + 2.72e3·34-s + 2.05e4·37-s − 1.41e4·38-s + 1.35e4·40-s + 8.81e3·41-s + 1.58e4·43-s − 9.16e3·44-s + 2.58e4·46-s + 3.39e4·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.948·5-s − 1.41·8-s + 1.34·10-s − 0.475·11-s + 0.621·13-s + 5/4·16-s − 0.285·17-s + 1.12·19-s − 1.42·20-s + 0.673·22-s − 1.27·23-s − 0.557·25-s − 0.879·26-s − 0.984·29-s − 0.372·31-s − 1.06·32-s + 0.403·34-s + 2.47·37-s − 1.58·38-s + 1.34·40-s + 0.818·41-s + 1.30·43-s − 0.713·44-s + 1.80·46-s + 2.23·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 53 T + 4552 T^{2} + 53 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 191 T + 24656 p T^{2} + 191 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 379 T + 488066 T^{2} - 379 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 20 p T + 1908514 T^{2} + 20 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1769 T + 2083758 T^{2} - 1769 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3236 T + 15452206 T^{2} + 3236 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4459 T + 37061638 T^{2} + 4459 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 1994 T - 6304813 T^{2} + 1994 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 20587 T + 238401006 T^{2} - 20587 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8814 T + 240678562 T^{2} - 8814 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 15853 T + 338678796 T^{2} - 15853 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 33912 T + 687783466 T^{2} - 33912 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 49239 T + 1320998110 T^{2} + 49239 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 56735 T + 2230528834 T^{2} + 56735 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 67508 T + 2826067262 T^{2} + 67508 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 75723 T + 3861149404 T^{2} - 75723 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8992 T - 681216182 T^{2} - 8992 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3201 T + 2311013380 T^{2} - 3201 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 26612 T + 3997648985 T^{2} - 26612 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 949 T + 367057696 T^{2} + 949 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 176562 T + 18855802834 T^{2} - 176562 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 129423 T + 11942811256 T^{2} - 129423 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.122967579408345301105672359664, −9.028836540491247939619092079475, −8.055112854525507532072218935418, −7.87488213768432862565109879127, −7.60495344669563393371368526228, −7.59986802901357946926291486359, −6.70346829860673560579288455765, −6.18638740186039916205990629628, −5.88171352231280897524639204479, −5.51679574437906724904096269878, −4.49425724696113673905834650185, −4.33068137825321978997266489929, −3.50311526236590679312837458846, −3.30090166998639905152420973040, −2.33876026178801805805615635408, −2.21931967658440275607589337776, −1.21058453152264209795029917407, −0.946722620384081625664183226747, 0, 0,
0.946722620384081625664183226747, 1.21058453152264209795029917407, 2.21931967658440275607589337776, 2.33876026178801805805615635408, 3.30090166998639905152420973040, 3.50311526236590679312837458846, 4.33068137825321978997266489929, 4.49425724696113673905834650185, 5.51679574437906724904096269878, 5.88171352231280897524639204479, 6.18638740186039916205990629628, 6.70346829860673560579288455765, 7.59986802901357946926291486359, 7.60495344669563393371368526228, 7.87488213768432862565109879127, 8.055112854525507532072218935418, 9.028836540491247939619092079475, 9.122967579408345301105672359664
