| L(s) = 1 | − 2-s − 4-s + 2·5-s + 3·8-s − 2·10-s − 4·11-s − 13-s − 16-s + 2·17-s − 2·20-s + 4·22-s − 25-s + 26-s + 10·29-s − 4·31-s − 5·32-s − 2·34-s − 2·37-s + 6·40-s + 6·41-s − 12·43-s + 4·44-s + 50-s + 52-s − 6·53-s − 8·55-s − 10·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 0.632·10-s − 1.20·11-s − 0.277·13-s − 1/4·16-s + 0.485·17-s − 0.447·20-s + 0.852·22-s − 1/5·25-s + 0.196·26-s + 1.85·29-s − 0.718·31-s − 0.883·32-s − 0.342·34-s − 0.328·37-s + 0.948·40-s + 0.937·41-s − 1.82·43-s + 0.603·44-s + 0.141·50-s + 0.138·52-s − 0.824·53-s − 1.07·55-s − 1.31·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.032632598223504972697717648194, −7.20789832531804473008179231275, −6.40781904393853776114096194211, −5.41348840769227002358378081555, −5.09771847455104642285437340033, −4.16792663500294091600799386995, −3.05843684806467631394197397430, −2.18396772418715587471415398313, −1.22246662492305573266308145199, 0,
1.22246662492305573266308145199, 2.18396772418715587471415398313, 3.05843684806467631394197397430, 4.16792663500294091600799386995, 5.09771847455104642285437340033, 5.41348840769227002358378081555, 6.40781904393853776114096194211, 7.20789832531804473008179231275, 8.032632598223504972697717648194
