L(s) = 1 | − 2-s + 4-s − 5·7-s − 8-s + 5·14-s + 16-s + 15·23-s − 7·25-s − 5·28-s − 2·31-s − 32-s − 12·41-s − 15·46-s + 3·47-s + 7·49-s + 7·50-s + 5·56-s + 2·62-s + 64-s − 9·71-s − 8·73-s − 11·79-s + 12·82-s − 9·89-s + 15·92-s − 3·94-s − 14·97-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.88·7-s − 0.353·8-s + 1.33·14-s + 1/4·16-s + 3.12·23-s − 7/5·25-s − 0.944·28-s − 0.359·31-s − 0.176·32-s − 1.87·41-s − 2.21·46-s + 0.437·47-s + 49-s + 0.989·50-s + 0.668·56-s + 0.254·62-s + 1/8·64-s − 1.06·71-s − 0.936·73-s − 1.23·79-s + 1.32·82-s − 0.953·89-s + 1.56·92-s − 0.309·94-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.419645368591443751810409784537, −8.904280917697275646853068856777, −8.652480179095890952277371090521, −7.88182478976990687669142588582, −7.15347676030799376791021350775, −6.99798914374157675830717731620, −6.49139062139326103439856603690, −5.87102828404298065596100724769, −5.36125293518532737464825972567, −4.58693033951185016739148820650, −3.64310443763390350744166738368, −3.16591465469024464891153500882, −2.65246982305560572298696734187, −1.40327001703361496701612763306, 0,
1.40327001703361496701612763306, 2.65246982305560572298696734187, 3.16591465469024464891153500882, 3.64310443763390350744166738368, 4.58693033951185016739148820650, 5.36125293518532737464825972567, 5.87102828404298065596100724769, 6.49139062139326103439856603690, 6.99798914374157675830717731620, 7.15347676030799376791021350775, 7.88182478976990687669142588582, 8.652480179095890952277371090521, 8.904280917697275646853068856777, 9.419645368591443751810409784537