| L(s) = 1 | − 2-s + 0.332·3-s + 4-s − 0.332·6-s − 3.51·7-s − 8-s − 2.88·9-s − 0.290·11-s + 0.332·12-s + 7.12·13-s + 3.51·14-s + 16-s − 6.17·17-s + 2.88·18-s + 5.83·19-s − 1.16·21-s + 0.290·22-s − 6.45·23-s − 0.332·24-s − 7.12·26-s − 1.96·27-s − 3.51·28-s − 1.18·29-s + 9.77·31-s − 32-s − 0.0965·33-s + 6.17·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.192·3-s + 0.5·4-s − 0.135·6-s − 1.32·7-s − 0.353·8-s − 0.963·9-s − 0.0874·11-s + 0.0961·12-s + 1.97·13-s + 0.938·14-s + 0.250·16-s − 1.49·17-s + 0.680·18-s + 1.33·19-s − 0.255·21-s + 0.0618·22-s − 1.34·23-s − 0.0679·24-s − 1.39·26-s − 0.377·27-s − 0.663·28-s − 0.219·29-s + 1.75·31-s − 0.176·32-s − 0.0168·33-s + 1.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9499251206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9499251206\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 0.332T + 3T^{2} \) |
| 7 | \( 1 + 3.51T + 7T^{2} \) |
| 11 | \( 1 + 0.290T + 11T^{2} \) |
| 13 | \( 1 - 7.12T + 13T^{2} \) |
| 17 | \( 1 + 6.17T + 17T^{2} \) |
| 19 | \( 1 - 5.83T + 19T^{2} \) |
| 23 | \( 1 + 6.45T + 23T^{2} \) |
| 29 | \( 1 + 1.18T + 29T^{2} \) |
| 31 | \( 1 - 9.77T + 31T^{2} \) |
| 41 | \( 1 - 1.64T + 41T^{2} \) |
| 43 | \( 1 + 5.34T + 43T^{2} \) |
| 47 | \( 1 - 5.69T + 47T^{2} \) |
| 53 | \( 1 - 9.32T + 53T^{2} \) |
| 59 | \( 1 - 5.94T + 59T^{2} \) |
| 61 | \( 1 + 1.27T + 61T^{2} \) |
| 67 | \( 1 - 6.04T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 1.71T + 73T^{2} \) |
| 79 | \( 1 - 8.25T + 79T^{2} \) |
| 83 | \( 1 - 2.41T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−9.147420528060077315825896886796, −8.565642540519832297451215346553, −7.944854279982176259349421047043, −6.70079999371230929419269797814, −6.29520190504385357396024301738, −5.52695826181187237218033393236, −3.94899526699099261482092665956, −3.23444949238114936540421407428, −2.27525741196793538135852558868, −0.70610509626668983854267367058,
0.70610509626668983854267367058, 2.27525741196793538135852558868, 3.23444949238114936540421407428, 3.94899526699099261482092665956, 5.52695826181187237218033393236, 6.29520190504385357396024301738, 6.70079999371230929419269797814, 7.944854279982176259349421047043, 8.565642540519832297451215346553, 9.147420528060077315825896886796
