| L(s) = 1 | + 32·2-s + 768·4-s + 1.63e4·8-s − 2.41e4·9-s − 5.52e4·11-s + 3.27e5·16-s − 7.71e5·18-s − 1.76e6·22-s − 1.07e6·23-s − 2.27e6·25-s − 5.18e6·29-s + 6.29e6·32-s − 1.85e7·36-s − 3.56e7·37-s − 1.75e7·43-s − 4.24e7·44-s − 3.42e7·46-s − 7.28e7·50-s − 1.66e8·53-s − 1.65e8·58-s + 1.17e8·64-s − 3.70e8·67-s + 7.40e8·71-s − 3.94e8·72-s − 1.13e9·74-s − 1.03e9·79-s + 1.93e8·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 1.22·9-s − 1.13·11-s + 5/4·16-s − 1.73·18-s − 1.61·22-s − 0.798·23-s − 1.16·25-s − 1.36·29-s + 1.06·32-s − 1.83·36-s − 3.12·37-s − 0.781·43-s − 1.70·44-s − 1.12·46-s − 1.64·50-s − 2.90·53-s − 1.92·58-s + 7/8·64-s − 2.24·67-s + 3.45·71-s − 1.73·72-s − 4.41·74-s − 3.00·79-s + 0.499·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2678 p^{2} T^{2} + p^{18} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2276394 T^{2} + p^{18} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 27644 T + p^{9} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 14268069562 T^{2} + p^{18} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 69999974750 T^{2} + p^{18} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 1460888662 p^{2} T^{2} + p^{18} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 535704 T + p^{9} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2591006 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 15702946511806 T^{2} + p^{18} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 17807910 T + p^{9} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 277548554067374 T^{2} + p^{18} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8759756 T + p^{9} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 1494446625586398 T^{2} + p^{18} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 83488490 T + p^{9} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 10675992491816982 T^{2} + p^{18} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 20903844502528282 T^{2} + p^{18} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 185492924 T + p^{9} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 370270672 T + p^{9} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 57221987368552210 T^{2} + p^{18} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 519907256 T + p^{9} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 284545962447299782 T^{2} + p^{18} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 612190546070247474 T^{2} + p^{18} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 910917015153568834 T^{2} + p^{18} T^{4} \) |
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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
−11.82236854078292406429467246080, −11.57756747628232037907336825579, −10.84923304428285637221471516830, −10.55728209305113004763534349412, −9.815060782048062711010606091129, −9.142642452237495937286932691175, −8.169995339787294510061760607715, −8.048047927829842809235348544844, −7.19596907989946325593263786520, −6.60159460989186824240392611696, −5.68921528380716236144615399985, −5.62952361146217294659237088467, −4.94922883066078126700334948675, −4.18311410678382181408925229707, −3.29512056835985570727065148396, −3.11480563363481081390916390525, −2.04457507436364048465701793830, −1.72991262628657713484677009694, 0, 0,
1.72991262628657713484677009694, 2.04457507436364048465701793830, 3.11480563363481081390916390525, 3.29512056835985570727065148396, 4.18311410678382181408925229707, 4.94922883066078126700334948675, 5.62952361146217294659237088467, 5.68921528380716236144615399985, 6.60159460989186824240392611696, 7.19596907989946325593263786520, 8.048047927829842809235348544844, 8.169995339787294510061760607715, 9.142642452237495937286932691175, 9.815060782048062711010606091129, 10.55728209305113004763534349412, 10.84923304428285637221471516830, 11.57756747628232037907336825579, 11.82236854078292406429467246080
