| L(s) = 1 | + 1.59·2-s + 0.546·4-s − 1.23·5-s + 2.46·7-s − 2.31·8-s − 1.97·10-s − 4.04·11-s + 6.08·13-s + 3.93·14-s − 4.79·16-s + 5.58·17-s − 0.469·19-s − 0.676·20-s − 6.44·22-s + 23-s − 3.46·25-s + 9.70·26-s + 1.34·28-s − 29-s − 2.00·31-s − 3.01·32-s + 8.90·34-s − 3.04·35-s + 5.90·37-s − 0.749·38-s + 2.86·40-s − 8.01·41-s + ⋯ |
| L(s) = 1 | + 1.12·2-s + 0.273·4-s − 0.553·5-s + 0.931·7-s − 0.820·8-s − 0.624·10-s − 1.21·11-s + 1.68·13-s + 1.05·14-s − 1.19·16-s + 1.35·17-s − 0.107·19-s − 0.151·20-s − 1.37·22-s + 0.208·23-s − 0.693·25-s + 1.90·26-s + 0.254·28-s − 0.185·29-s − 0.359·31-s − 0.532·32-s + 1.52·34-s − 0.515·35-s + 0.970·37-s − 0.121·38-s + 0.453·40-s − 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.162930715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.162930715\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.59T + 2T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 - 6.08T + 13T^{2} \) |
| 17 | \( 1 - 5.58T + 17T^{2} \) |
| 19 | \( 1 + 0.469T + 19T^{2} \) |
| 31 | \( 1 + 2.00T + 31T^{2} \) |
| 37 | \( 1 - 5.90T + 37T^{2} \) |
| 41 | \( 1 + 8.01T + 41T^{2} \) |
| 43 | \( 1 - 6.87T + 43T^{2} \) |
| 47 | \( 1 - 8.37T + 47T^{2} \) |
| 53 | \( 1 + 2.29T + 53T^{2} \) |
| 59 | \( 1 + 6.71T + 59T^{2} \) |
| 61 | \( 1 - 8.00T + 61T^{2} \) |
| 67 | \( 1 - 6.11T + 67T^{2} \) |
| 71 | \( 1 + 0.713T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 2.01T + 79T^{2} \) |
| 83 | \( 1 - 7.66T + 83T^{2} \) |
| 89 | \( 1 + 0.733T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.025647073609356724652875137245, −7.49293176048396843117901382186, −6.38010949345553253650771924576, −5.65523196086918347016544708238, −5.25014450299923618651589087474, −4.41426914351454923058811243509, −3.72235048920373515569966553561, −3.15642902133962800183148083515, −2.06201343701814348748418044618, −0.798292542091018594733436809285,
0.798292542091018594733436809285, 2.06201343701814348748418044618, 3.15642902133962800183148083515, 3.72235048920373515569966553561, 4.41426914351454923058811243509, 5.25014450299923618651589087474, 5.65523196086918347016544708238, 6.38010949345553253650771924576, 7.49293176048396843117901382186, 8.025647073609356724652875137245
