| L(s) = 1 | − 2-s + 4-s − 3.28·5-s + 3.80·7-s − 8-s + 3.28·10-s − 1.17·11-s − 13-s − 3.80·14-s + 16-s + 1.50·17-s − 2.81·19-s − 3.28·20-s + 1.17·22-s − 5.32·23-s + 5.76·25-s + 26-s + 3.80·28-s + 8.71·29-s − 31-s − 32-s − 1.50·34-s − 12.4·35-s − 1.41·37-s + 2.81·38-s + 3.28·40-s + 0.248·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.46·5-s + 1.43·7-s − 0.353·8-s + 1.03·10-s − 0.354·11-s − 0.277·13-s − 1.01·14-s + 0.250·16-s + 0.365·17-s − 0.646·19-s − 0.733·20-s + 0.250·22-s − 1.11·23-s + 1.15·25-s + 0.196·26-s + 0.718·28-s + 1.61·29-s − 0.179·31-s − 0.176·32-s − 0.258·34-s − 2.10·35-s − 0.233·37-s + 0.457·38-s + 0.518·40-s + 0.0388·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9486187285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9486187285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 5 | \( 1 + 3.28T + 5T^{2} \) |
| 7 | \( 1 - 3.80T + 7T^{2} \) |
| 11 | \( 1 + 1.17T + 11T^{2} \) |
| 17 | \( 1 - 1.50T + 17T^{2} \) |
| 19 | \( 1 + 2.81T + 19T^{2} \) |
| 23 | \( 1 + 5.32T + 23T^{2} \) |
| 29 | \( 1 - 8.71T + 29T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 - 0.248T + 41T^{2} \) |
| 43 | \( 1 - 5.84T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 9.55T + 53T^{2} \) |
| 59 | \( 1 + 9.72T + 59T^{2} \) |
| 61 | \( 1 - 3.35T + 61T^{2} \) |
| 67 | \( 1 + 8.28T + 67T^{2} \) |
| 71 | \( 1 + 4.28T + 71T^{2} \) |
| 73 | \( 1 - 7.52T + 73T^{2} \) |
| 79 | \( 1 + 6.10T + 79T^{2} \) |
| 83 | \( 1 - 6.71T + 83T^{2} \) |
| 89 | \( 1 - 9.00T + 89T^{2} \) |
| 97 | \( 1 - 5.07T + 97T^{2} \) |
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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
−7.898513679751386059561226638739, −7.61805924857647301375124116444, −6.78833597309020700687365158424, −5.89333159533056977715892954479, −4.88385789389135741225352779794, −4.42044684815786878891911379854, −3.57861127288059118632964319244, −2.57935208225269790206616266618, −1.64666749413372134955195651283, −0.55612914486715409600800882835,
0.55612914486715409600800882835, 1.64666749413372134955195651283, 2.57935208225269790206616266618, 3.57861127288059118632964319244, 4.42044684815786878891911379854, 4.88385789389135741225352779794, 5.89333159533056977715892954479, 6.78833597309020700687365158424, 7.61805924857647301375124116444, 7.898513679751386059561226638739
