L(s) = 1 | + 1.68·2-s + 3-s + 0.841·4-s + 1.68·6-s − 4.79·7-s − 1.95·8-s + 9-s + 4.98·11-s + 0.841·12-s + 3.91·13-s − 8.08·14-s − 4.97·16-s − 6.51·17-s + 1.68·18-s − 4.65·19-s − 4.79·21-s + 8.40·22-s + 1.62·23-s − 1.95·24-s + 6.59·26-s + 27-s − 4.03·28-s + 3.76·29-s − 7.29·31-s − 4.47·32-s + 4.98·33-s − 10.9·34-s + ⋯ |
L(s) = 1 | + 1.19·2-s + 0.577·3-s + 0.420·4-s + 0.688·6-s − 1.81·7-s − 0.690·8-s + 0.333·9-s + 1.50·11-s + 0.242·12-s + 1.08·13-s − 2.16·14-s − 1.24·16-s − 1.57·17-s + 0.397·18-s − 1.06·19-s − 1.04·21-s + 1.79·22-s + 0.339·23-s − 0.398·24-s + 1.29·26-s + 0.192·27-s − 0.762·28-s + 0.699·29-s − 1.31·31-s − 0.791·32-s + 0.868·33-s − 1.88·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 - 1.68T + 2T^{2} \) |
| 7 | \( 1 + 4.79T + 7T^{2} \) |
| 11 | \( 1 - 4.98T + 11T^{2} \) |
| 13 | \( 1 - 3.91T + 13T^{2} \) |
| 17 | \( 1 + 6.51T + 17T^{2} \) |
| 19 | \( 1 + 4.65T + 19T^{2} \) |
| 23 | \( 1 - 1.62T + 23T^{2} \) |
| 29 | \( 1 - 3.76T + 29T^{2} \) |
| 31 | \( 1 + 7.29T + 31T^{2} \) |
| 37 | \( 1 + 8.07T + 37T^{2} \) |
| 41 | \( 1 + 6.49T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 53 | \( 1 + 2.23T + 53T^{2} \) |
| 59 | \( 1 + 2.49T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 - 6.18T + 67T^{2} \) |
| 71 | \( 1 - 9.54T + 71T^{2} \) |
| 73 | \( 1 - 4.36T + 73T^{2} \) |
| 79 | \( 1 - 3.21T + 79T^{2} \) |
| 83 | \( 1 + 5.71T + 83T^{2} \) |
| 89 | \( 1 + 1.60T + 89T^{2} \) |
| 97 | \( 1 - 8.26T + 97T^{2} \) |
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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
−8.537305809126125293963175107499, −6.88609774868796798910628427357, −6.55028640750459644382139487666, −6.20723504525777995257010444854, −5.00221149780268878701081449350, −3.97834583410543727237989832286, −3.68920749053424735481698602283, −2.97430301927446834008143164450, −1.82245990115195922441416620902, 0,
1.82245990115195922441416620902, 2.97430301927446834008143164450, 3.68920749053424735481698602283, 3.97834583410543727237989832286, 5.00221149780268878701081449350, 6.20723504525777995257010444854, 6.55028640750459644382139487666, 6.88609774868796798910628427357, 8.537305809126125293963175107499