| L(s) = 1 | − 2-s − 2.54·3-s + 4-s + 2.54·6-s + 7-s − 8-s + 3.46·9-s − 5.72·11-s − 2.54·12-s + 3.48·13-s − 14-s + 16-s + 17-s − 3.46·18-s − 4.55·19-s − 2.54·21-s + 5.72·22-s + 8.26·23-s + 2.54·24-s − 3.48·26-s − 1.17·27-s + 28-s − 0.990·29-s − 3.99·31-s − 32-s + 14.5·33-s − 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.46·3-s + 0.5·4-s + 1.03·6-s + 0.377·7-s − 0.353·8-s + 1.15·9-s − 1.72·11-s − 0.733·12-s + 0.965·13-s − 0.267·14-s + 0.250·16-s + 0.242·17-s − 0.815·18-s − 1.04·19-s − 0.554·21-s + 1.22·22-s + 1.72·23-s + 0.518·24-s − 0.683·26-s − 0.225·27-s + 0.188·28-s − 0.183·29-s − 0.716·31-s − 0.176·32-s + 2.53·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5281460006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5281460006\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2.54T + 3T^{2} \) |
| 11 | \( 1 + 5.72T + 11T^{2} \) |
| 13 | \( 1 - 3.48T + 13T^{2} \) |
| 19 | \( 1 + 4.55T + 19T^{2} \) |
| 23 | \( 1 - 8.26T + 23T^{2} \) |
| 29 | \( 1 + 0.990T + 29T^{2} \) |
| 31 | \( 1 + 3.99T + 31T^{2} \) |
| 37 | \( 1 + 1.96T + 37T^{2} \) |
| 41 | \( 1 - 7.88T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 8.49T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 0.952T + 61T^{2} \) |
| 67 | \( 1 - 3.87T + 67T^{2} \) |
| 71 | \( 1 + 6.23T + 71T^{2} \) |
| 73 | \( 1 - 7.07T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−8.084666491451533052619734916059, −7.36750238390498623716899242447, −6.61153051407892740931434989897, −6.03296969396760066019084643604, −5.16209401821585707075252533599, −4.93820228599042993735403284625, −3.63670007026117245088881599510, −2.58905055589431417096630079642, −1.50231262080674226777843158888, −0.47643358674089987847698833187,
0.47643358674089987847698833187, 1.50231262080674226777843158888, 2.58905055589431417096630079642, 3.63670007026117245088881599510, 4.93820228599042993735403284625, 5.16209401821585707075252533599, 6.03296969396760066019084643604, 6.61153051407892740931434989897, 7.36750238390498623716899242447, 8.084666491451533052619734916059
