| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·5-s − 2·6-s + 8-s + 9-s − 2·10-s + 11-s − 2·12-s + 2·13-s + 4·15-s + 16-s + 18-s − 2·19-s − 2·20-s + 22-s − 2·24-s − 25-s + 2·26-s + 4·27-s + 6·29-s + 4·30-s + 4·31-s + 32-s − 2·33-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.577·12-s + 0.554·13-s + 1.03·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s − 0.447·20-s + 0.213·22-s − 0.408·24-s − 1/5·25-s + 0.392·26-s + 0.769·27-s + 1.11·29-s + 0.730·30-s + 0.718·31-s + 0.176·32-s − 0.348·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.380256278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.380256278\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.28233411665073803151231254247, −8.999428877443585420772298249849, −8.047710162724936352024101809585, −7.16363172811847616280629421186, −6.24069004878374419814700011775, −5.71863635665082255443003600094, −4.55231601613734394095039873232, −4.01218149796097302444856169190, −2.70614041997722497205323328529, −0.868364880052939490205998305824,
0.868364880052939490205998305824, 2.70614041997722497205323328529, 4.01218149796097302444856169190, 4.55231601613734394095039873232, 5.71863635665082255443003600094, 6.24069004878374419814700011775, 7.16363172811847616280629421186, 8.047710162724936352024101809585, 8.999428877443585420772298249849, 10.28233411665073803151231254247
