| L(s) = 1 | + 2-s + 1.44·3-s + 4-s + 3.26·5-s + 1.44·6-s − 3.72·7-s + 8-s − 0.903·9-s + 3.26·10-s − 4.09·11-s + 1.44·12-s − 2.98·13-s − 3.72·14-s + 4.72·15-s + 16-s − 1.49·17-s − 0.903·18-s − 1.05·19-s + 3.26·20-s − 5.39·21-s − 4.09·22-s − 23-s + 1.44·24-s + 5.64·25-s − 2.98·26-s − 5.65·27-s − 3.72·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.836·3-s + 0.5·4-s + 1.45·5-s + 0.591·6-s − 1.40·7-s + 0.353·8-s − 0.301·9-s + 1.03·10-s − 1.23·11-s + 0.418·12-s − 0.826·13-s − 0.996·14-s + 1.21·15-s + 0.250·16-s − 0.362·17-s − 0.212·18-s − 0.241·19-s + 0.729·20-s − 1.17·21-s − 0.873·22-s − 0.208·23-s + 0.295·24-s + 1.12·25-s − 0.584·26-s − 1.08·27-s − 0.704·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 5 | \( 1 - 3.26T + 5T^{2} \) |
| 7 | \( 1 + 3.72T + 7T^{2} \) |
| 11 | \( 1 + 4.09T + 11T^{2} \) |
| 13 | \( 1 + 2.98T + 13T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 19 | \( 1 + 1.05T + 19T^{2} \) |
| 29 | \( 1 + 4.90T + 29T^{2} \) |
| 31 | \( 1 - 6.60T + 31T^{2} \) |
| 37 | \( 1 + 8.08T + 37T^{2} \) |
| 41 | \( 1 - 2.30T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 + 8.29T + 47T^{2} \) |
| 53 | \( 1 - 0.309T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 7.82T + 71T^{2} \) |
| 73 | \( 1 - 4.03T + 73T^{2} \) |
| 79 | \( 1 + 2.73T + 79T^{2} \) |
| 83 | \( 1 + 0.859T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 2.53T + 97T^{2} \) |
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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
−7.70265310246439075850680400276, −6.71856333728085143324346750009, −6.36288309738259872586576872841, −5.43454636933089177738709184273, −5.11307359650290367844426231017, −3.83219252643360713110794570104, −3.00615383741542121194122857491, −2.53392538418888226581595060298, −1.92150845968102997895711877001, 0,
1.92150845968102997895711877001, 2.53392538418888226581595060298, 3.00615383741542121194122857491, 3.83219252643360713110794570104, 5.11307359650290367844426231017, 5.43454636933089177738709184273, 6.36288309738259872586576872841, 6.71856333728085143324346750009, 7.70265310246439075850680400276
