| L(s) = 1 | − 2·5-s − 7-s − 3·9-s − 4·11-s − 6·13-s + 6·17-s − 4·19-s + 23-s − 25-s − 29-s − 4·31-s + 2·35-s − 6·37-s − 6·41-s + 4·43-s + 6·45-s + 4·47-s + 49-s − 6·53-s + 8·55-s + 8·59-s + 6·61-s + 3·63-s + 12·65-s + 4·67-s + 8·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 9-s − 1.20·11-s − 1.66·13-s + 1.45·17-s − 0.917·19-s + 0.208·23-s − 1/5·25-s − 0.185·29-s − 0.718·31-s + 0.338·35-s − 0.986·37-s − 0.937·41-s + 0.609·43-s + 0.894·45-s + 0.583·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s + 1.04·59-s + 0.768·61-s + 0.377·63-s + 1.48·65-s + 0.488·67-s + 0.949·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.87227787979353, −12.68154458165499, −12.21466691268977, −11.86911114837686, −11.38231516649186, −10.84117461723515, −10.41136079765858, −9.999828804561629, −9.520923915182494, −9.006674955391995, −8.318016077565988, −8.076096282396493, −7.651879123984317, −7.140489706909981, −6.788686650512229, −5.959943783986160, −5.461208753815225, −5.195876218084414, −4.670587480788298, −3.907913409470123, −3.462005538485095, −2.984781321193640, −2.388146126715771, −2.011433277933604, −0.8627894009869240, 0, 0,
0.8627894009869240, 2.011433277933604, 2.388146126715771, 2.984781321193640, 3.462005538485095, 3.907913409470123, 4.670587480788298, 5.195876218084414, 5.461208753815225, 5.959943783986160, 6.788686650512229, 7.140489706909981, 7.651879123984317, 8.076096282396493, 8.318016077565988, 9.006674955391995, 9.520923915182494, 9.999828804561629, 10.41136079765858, 10.84117461723515, 11.38231516649186, 11.86911114837686, 12.21466691268977, 12.68154458165499, 12.87227787979353
