| L(s) = 1 | − 4-s + 2·9-s − 2·11-s + 16-s − 4·19-s + 16·31-s − 2·36-s + 24·41-s + 2·44-s − 49-s + 4·61-s − 64-s + 24·71-s + 4·76-s − 28·79-s − 5·81-s − 12·89-s − 4·99-s − 12·101-s + 32·109-s + 3·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/3·9-s − 0.603·11-s + 1/4·16-s − 0.917·19-s + 2.87·31-s − 1/3·36-s + 3.74·41-s + 0.301·44-s − 1/7·49-s + 0.512·61-s − 1/8·64-s + 2.84·71-s + 0.458·76-s − 3.15·79-s − 5/9·81-s − 1.27·89-s − 0.402·99-s − 1.19·101-s + 3.06·109-s + 3/11·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.411763205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.411763205\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.016160188086369078198345489357, −8.352863938509922763545728674005, −7.991962657600809798209103536536, −7.55238501891779828730565396992, −7.36133733695809529319806056749, −6.76400685935591825903254017260, −6.44394220018345693556867022891, −6.12696124048147765798748698660, −5.67373125888953515517706383635, −5.33912951447419906957897177077, −4.71851625510094114873538830345, −4.52558211100606653907995824683, −4.03421710480520412941972240603, −3.97978197307482576611617739318, −3.07883553678661024807136028429, −2.60539277417018856901612260098, −2.49395615683054007024966929769, −1.66397717110913711662857335883, −1.00682387342408775416500634629, −0.54841090217872995800557608469,
0.54841090217872995800557608469, 1.00682387342408775416500634629, 1.66397717110913711662857335883, 2.49395615683054007024966929769, 2.60539277417018856901612260098, 3.07883553678661024807136028429, 3.97978197307482576611617739318, 4.03421710480520412941972240603, 4.52558211100606653907995824683, 4.71851625510094114873538830345, 5.33912951447419906957897177077, 5.67373125888953515517706383635, 6.12696124048147765798748698660, 6.44394220018345693556867022891, 6.76400685935591825903254017260, 7.36133733695809529319806056749, 7.55238501891779828730565396992, 7.991962657600809798209103536536, 8.352863938509922763545728674005, 9.016160188086369078198345489357
