| L(s) = 1 | − 3·2-s + 4·4-s − 3·8-s − 9-s − 6·11-s − 2·13-s + 3·16-s − 6·17-s + 3·18-s + 6·19-s + 18·22-s − 12·23-s − 5·25-s + 6·26-s + 10·31-s − 6·32-s + 18·34-s − 4·36-s − 4·37-s − 18·38-s − 16·43-s − 24·44-s + 36·46-s − 6·47-s + 15·50-s − 8·52-s − 6·53-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s − 1.06·8-s − 1/3·9-s − 1.80·11-s − 0.554·13-s + 3/4·16-s − 1.45·17-s + 0.707·18-s + 1.37·19-s + 3.83·22-s − 2.50·23-s − 25-s + 1.17·26-s + 1.79·31-s − 1.06·32-s + 3.08·34-s − 2/3·36-s − 0.657·37-s − 2.91·38-s − 2.43·43-s − 3.61·44-s + 5.30·46-s − 0.875·47-s + 2.12·50-s − 1.10·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 95 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 107 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 173 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−10.04739722696444408909984293125, −9.857629142563251110834527806246, −9.696184314450238948679944456992, −9.129976788151655705340675717080, −8.391231870394862979874925174825, −8.275967704225302042046045152173, −7.973177014790747387573699419633, −7.69159979186934493867487892216, −6.99636728548442572942115661332, −6.53198305530027523178787819232, −6.00430706481802488805935210728, −5.29432129631228304703014190482, −5.05312515101241452638497988149, −4.25717014941578661961878362485, −3.50247610759214200813467242100, −2.72343313377748065226121554450, −2.24242961659627172464937550509, −1.48329437481324086898099118124, 0, 0,
1.48329437481324086898099118124, 2.24242961659627172464937550509, 2.72343313377748065226121554450, 3.50247610759214200813467242100, 4.25717014941578661961878362485, 5.05312515101241452638497988149, 5.29432129631228304703014190482, 6.00430706481802488805935210728, 6.53198305530027523178787819232, 6.99636728548442572942115661332, 7.69159979186934493867487892216, 7.973177014790747387573699419633, 8.275967704225302042046045152173, 8.391231870394862979874925174825, 9.129976788151655705340675717080, 9.696184314450238948679944456992, 9.857629142563251110834527806246, 10.04739722696444408909984293125
