| L(s) = 1 | − 2·4-s + 2·7-s + 9-s − 2·11-s + 4·16-s + 2·23-s − 4·28-s + 4·29-s − 2·36-s − 2·37-s + 14·43-s + 4·44-s − 3·49-s − 8·53-s + 2·63-s − 8·64-s + 8·67-s − 24·71-s − 4·77-s − 8·79-s − 8·81-s − 4·92-s − 2·99-s + 16·107-s − 24·109-s + 8·112-s − 6·113-s + ⋯ |
| L(s) = 1 | − 4-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 16-s + 0.417·23-s − 0.755·28-s + 0.742·29-s − 1/3·36-s − 0.328·37-s + 2.13·43-s + 0.603·44-s − 3/7·49-s − 1.09·53-s + 0.251·63-s − 64-s + 0.977·67-s − 2.84·71-s − 0.455·77-s − 0.900·79-s − 8/9·81-s − 0.417·92-s − 0.201·99-s + 1.54·107-s − 2.29·109-s + 0.755·112-s − 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
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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
−7.63348729974613337925505720358, −7.33814355294077068602873323503, −6.72277750243453825606179562032, −6.19480920810233323131249637324, −5.74548836625251696018218276721, −5.30062003638913056307326926019, −4.93638619313269311825207708877, −4.36155028925617016669924135513, −4.28142100186073480005823941465, −3.53294224684022494246624725994, −2.94493941314449159963107340310, −2.47488423943830886280853592741, −1.56639758224286743700678377480, −1.06662827284124857085751865504, 0,
1.06662827284124857085751865504, 1.56639758224286743700678377480, 2.47488423943830886280853592741, 2.94493941314449159963107340310, 3.53294224684022494246624725994, 4.28142100186073480005823941465, 4.36155028925617016669924135513, 4.93638619313269311825207708877, 5.30062003638913056307326926019, 5.74548836625251696018218276721, 6.19480920810233323131249637324, 6.72277750243453825606179562032, 7.33814355294077068602873323503, 7.63348729974613337925505720358
