| L(s) = 1 | − 4-s + 5-s + 9-s − 11-s + 16-s − 20-s + 23-s − 3·25-s − 3·31-s − 36-s + 37-s + 44-s + 45-s − 15·47-s + 49-s + 53-s − 55-s + 59-s − 64-s + 9·71-s + 80-s + 81-s − 11·89-s − 92-s + 24·97-s − 99-s + 3·100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 1/4·16-s − 0.223·20-s + 0.208·23-s − 3/5·25-s − 0.538·31-s − 1/6·36-s + 0.164·37-s + 0.150·44-s + 0.149·45-s − 2.18·47-s + 1/7·49-s + 0.137·53-s − 0.134·55-s + 0.130·59-s − 1/8·64-s + 1.06·71-s + 0.111·80-s + 1/9·81-s − 1.16·89-s − 0.104·92-s + 2.43·97-s − 0.100·99-s + 3/10·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 135 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.45179584550313776999997863292, −7.11225607135635514742408159461, −6.60248673433595561072935718912, −6.09542947843444896540059599698, −5.86885160376216428465346303013, −5.20806518568034195442740866619, −4.91634583716612964186305001259, −4.54252907369302845515223785330, −3.83701979890853209393933480235, −3.53695634471794134164318268570, −2.94442257014482982611578277167, −2.24390920759158476069795190202, −1.77265863666683343110776821285, −1.01317221725492991929464572058, 0,
1.01317221725492991929464572058, 1.77265863666683343110776821285, 2.24390920759158476069795190202, 2.94442257014482982611578277167, 3.53695634471794134164318268570, 3.83701979890853209393933480235, 4.54252907369302845515223785330, 4.91634583716612964186305001259, 5.20806518568034195442740866619, 5.86885160376216428465346303013, 6.09542947843444896540059599698, 6.60248673433595561072935718912, 7.11225607135635514742408159461, 7.45179584550313776999997863292
