| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 7-s + 8-s + 9-s − 1.46·11-s + 2·12-s − 1.26·13-s + 14-s + 16-s − 3.46·17-s + 18-s + 7.46·19-s + 2·21-s − 1.46·22-s + 6·23-s + 2·24-s − 5·25-s − 1.26·26-s − 4·27-s + 28-s + 0.535·29-s + 5.46·31-s + 32-s − 2.92·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.5·4-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.441·11-s + 0.577·12-s − 0.351·13-s + 0.267·14-s + 0.250·16-s − 0.840·17-s + 0.235·18-s + 1.71·19-s + 0.436·21-s − 0.312·22-s + 1.25·23-s + 0.408·24-s − 25-s − 0.248·26-s − 0.769·27-s + 0.188·28-s + 0.0995·29-s + 0.981·31-s + 0.176·32-s − 0.509·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.140210010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.140210010\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 7.46T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 0.535T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 - 8.73T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 5.66T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 + 2.73T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−7.901656456759484453243606721129, −7.53135419003607527305076287689, −6.76800400786557748969252683145, −5.76644517276678740487189763259, −5.14480992510394346940163654046, −4.34206649402798819291373903264, −3.58947919687222760896454807105, −2.69170318201037302977589338394, −2.36276462851237237602943438730, −1.04769285250927742967463515245,
1.04769285250927742967463515245, 2.36276462851237237602943438730, 2.69170318201037302977589338394, 3.58947919687222760896454807105, 4.34206649402798819291373903264, 5.14480992510394346940163654046, 5.76644517276678740487189763259, 6.76800400786557748969252683145, 7.53135419003607527305076287689, 7.901656456759484453243606721129
