L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 5·9-s + 6·11-s + 5·16-s + 10·18-s − 12·22-s + 2·25-s + 6·29-s − 6·32-s − 15·36-s − 2·37-s − 8·43-s + 18·44-s − 4·50-s − 12·58-s + 7·64-s + 4·67-s − 6·71-s + 20·72-s + 4·74-s + 25·79-s + 16·81-s + 16·86-s − 24·88-s − 30·99-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 5/3·9-s + 1.80·11-s + 5/4·16-s + 2.35·18-s − 2.55·22-s + 2/5·25-s + 1.11·29-s − 1.06·32-s − 5/2·36-s − 0.328·37-s − 1.21·43-s + 2.71·44-s − 0.565·50-s − 1.57·58-s + 7/8·64-s + 0.488·67-s − 0.712·71-s + 2.35·72-s + 0.464·74-s + 2.81·79-s + 16/9·81-s + 1.72·86-s − 2.55·88-s − 3.01·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8705330231\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8705330231\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 143 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
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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
−8.248518264204835053535772531099, −7.79962200523138252853113049258, −7.20507315988102608870866437335, −6.68520823045857904079903031451, −6.42562872678811285539295855882, −6.17474235583500945418451460037, −5.46717997811362667477862847034, −5.08415141677451039205800553710, −4.36149377738493704418872680385, −3.64113163695606166841995590200, −3.25213177801651800178143470021, −2.67971661503469679878583532858, −2.03102507827915237522854626797, −1.31918947733807367523596345875, −0.56440009517777295281733284422,
0.56440009517777295281733284422, 1.31918947733807367523596345875, 2.03102507827915237522854626797, 2.67971661503469679878583532858, 3.25213177801651800178143470021, 3.64113163695606166841995590200, 4.36149377738493704418872680385, 5.08415141677451039205800553710, 5.46717997811362667477862847034, 6.17474235583500945418451460037, 6.42562872678811285539295855882, 6.68520823045857904079903031451, 7.20507315988102608870866437335, 7.79962200523138252853113049258, 8.248518264204835053535772531099