| L(s) = 1 | + 4·4-s + 5·9-s − 6·11-s + 12·16-s − 4·19-s − 6·29-s − 8·31-s + 20·36-s − 24·41-s − 24·44-s − 49-s + 16·61-s + 32·64-s − 16·76-s + 2·79-s + 16·81-s + 24·89-s − 30·99-s + 12·101-s + 14·109-s − 24·116-s + 5·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2·4-s + 5/3·9-s − 1.80·11-s + 3·16-s − 0.917·19-s − 1.11·29-s − 1.43·31-s + 10/3·36-s − 3.74·41-s − 3.61·44-s − 1/7·49-s + 2.04·61-s + 4·64-s − 1.83·76-s + 0.225·79-s + 16/9·81-s + 2.54·89-s − 3.01·99-s + 1.19·101-s + 1.34·109-s − 2.22·116-s + 5/11·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.944888103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944888103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 193 T^{2} + p^{2} T^{4} \) |
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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
−12.86318801592704762278195545106, −12.67384449893208941937989550106, −11.71056418237357816472976395050, −11.67418408201091377603933954843, −10.69582023742138564708576503683, −10.69041874857904440775009233123, −10.08317058973632407671778685513, −9.898878392443386271295467110125, −8.815854534325350027084581673107, −8.147871851626506410832709692863, −7.54287668023440162427597917123, −7.37276997896353354604911032312, −6.69110116830137008686653654391, −6.36652082627462060432782413242, −5.32968870040299501719963145264, −5.12657998933874325446277157686, −3.84327066302090326621443783466, −3.26198202634456802309882085776, −2.15216858732523752878913131852, −1.81420037246371677867399420653,
1.81420037246371677867399420653, 2.15216858732523752878913131852, 3.26198202634456802309882085776, 3.84327066302090326621443783466, 5.12657998933874325446277157686, 5.32968870040299501719963145264, 6.36652082627462060432782413242, 6.69110116830137008686653654391, 7.37276997896353354604911032312, 7.54287668023440162427597917123, 8.147871851626506410832709692863, 8.815854534325350027084581673107, 9.898878392443386271295467110125, 10.08317058973632407671778685513, 10.69041874857904440775009233123, 10.69582023742138564708576503683, 11.67418408201091377603933954843, 11.71056418237357816472976395050, 12.67384449893208941937989550106, 12.86318801592704762278195545106
