| L(s) = 1 | − 2-s − 4-s + 4·7-s + 3·8-s − 9-s + 8·11-s − 4·14-s − 16-s + 18-s − 8·22-s + 8·23-s − 25-s − 4·28-s − 5·32-s + 36-s + 8·37-s − 8·44-s − 8·46-s + 9·49-s + 50-s + 8·53-s + 12·56-s − 4·63-s + 7·64-s − 3·72-s − 8·74-s + 32·77-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.51·7-s + 1.06·8-s − 1/3·9-s + 2.41·11-s − 1.06·14-s − 1/4·16-s + 0.235·18-s − 1.70·22-s + 1.66·23-s − 1/5·25-s − 0.755·28-s − 0.883·32-s + 1/6·36-s + 1.31·37-s − 1.20·44-s − 1.17·46-s + 9/7·49-s + 0.141·50-s + 1.09·53-s + 1.60·56-s − 0.503·63-s + 7/8·64-s − 0.353·72-s − 0.929·74-s + 3.64·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.907711219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.907711219\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.395588398684051700253522250398, −8.003898044235172358258455233144, −7.52574597566296120465702658855, −7.10462812424705825434021419505, −6.59557190913696964434674840198, −6.19775834822611601872413024051, −5.38359673457084232872540127643, −5.14779113402398772275184331303, −4.50307703178478698619428447617, −4.10383889494897225786069408110, −3.76614710718475734091024708387, −2.87446886218633013135158275759, −2.00818875239307364862621206639, −1.28760838012133746273806980988, −0.962913415112682620200246474198,
0.962913415112682620200246474198, 1.28760838012133746273806980988, 2.00818875239307364862621206639, 2.87446886218633013135158275759, 3.76614710718475734091024708387, 4.10383889494897225786069408110, 4.50307703178478698619428447617, 5.14779113402398772275184331303, 5.38359673457084232872540127643, 6.19775834822611601872413024051, 6.59557190913696964434674840198, 7.10462812424705825434021419505, 7.52574597566296120465702658855, 8.003898044235172358258455233144, 8.395588398684051700253522250398
