| L(s) = 1 | + 4·7-s − 4·17-s − 12·23-s − 8·25-s + 16·31-s − 16·47-s − 2·49-s − 20·71-s + 8·73-s − 8·89-s − 4·97-s − 12·103-s − 12·113-s − 16·119-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 48·161-s + 163-s + 167-s − 24·169-s + 173-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.970·17-s − 2.50·23-s − 8/5·25-s + 2.87·31-s − 2.33·47-s − 2/7·49-s − 2.37·71-s + 0.936·73-s − 0.847·89-s − 0.406·97-s − 1.18·103-s − 1.12·113-s − 1.46·119-s − 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 3.78·161-s + 0.0783·163-s + 0.0773·167-s − 1.84·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 84 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 164 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−7.51492538936526841986057630477, −7.48654113349428805147724410845, −6.73082787597525248820353617691, −6.45742913161854301633481661620, −6.16449417391641614736057879116, −6.03865837456829704952567991427, −5.26907252264224647150774383578, −5.24904918684148510310328686897, −4.58477741635161640209811755153, −4.52351514702833609690323614937, −4.05969940102495599980727480232, −3.87882403593986685800817551240, −3.19423179007541018340507254649, −2.78097724796515450927844404860, −2.21747932060423342208001006657, −2.05668401343670540566030108067, −1.38943510269200836088248952408, −1.29403526790407359603601570518, 0, 0,
1.29403526790407359603601570518, 1.38943510269200836088248952408, 2.05668401343670540566030108067, 2.21747932060423342208001006657, 2.78097724796515450927844404860, 3.19423179007541018340507254649, 3.87882403593986685800817551240, 4.05969940102495599980727480232, 4.52351514702833609690323614937, 4.58477741635161640209811755153, 5.24904918684148510310328686897, 5.26907252264224647150774383578, 6.03865837456829704952567991427, 6.16449417391641614736057879116, 6.45742913161854301633481661620, 6.73082787597525248820353617691, 7.48654113349428805147724410845, 7.51492538936526841986057630477
