| L(s) = 1 | − 2-s − 3.35·3-s + 4-s − 5-s + 3.35·6-s + 1.93·7-s − 8-s + 8.26·9-s + 10-s + 4.37·11-s − 3.35·12-s + 6.33·13-s − 1.93·14-s + 3.35·15-s + 16-s − 6.50·17-s − 8.26·18-s + 6.98·19-s − 20-s − 6.50·21-s − 4.37·22-s − 0.991·23-s + 3.35·24-s + 25-s − 6.33·26-s − 17.6·27-s + 1.93·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.93·3-s + 0.5·4-s − 0.447·5-s + 1.37·6-s + 0.732·7-s − 0.353·8-s + 2.75·9-s + 0.316·10-s + 1.31·11-s − 0.969·12-s + 1.75·13-s − 0.517·14-s + 0.866·15-s + 0.250·16-s − 1.57·17-s − 1.94·18-s + 1.60·19-s − 0.223·20-s − 1.41·21-s − 0.932·22-s − 0.206·23-s + 0.685·24-s + 0.200·25-s − 1.24·26-s − 3.40·27-s + 0.366·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 + 3.35T + 3T^{2} \) |
| 7 | \( 1 - 1.93T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 - 6.33T + 13T^{2} \) |
| 17 | \( 1 + 6.50T + 17T^{2} \) |
| 19 | \( 1 - 6.98T + 19T^{2} \) |
| 23 | \( 1 + 0.991T + 23T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 + 2.57T + 31T^{2} \) |
| 37 | \( 1 + 3.55T + 37T^{2} \) |
| 41 | \( 1 + 5.58T + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 + 3.58T + 47T^{2} \) |
| 53 | \( 1 + 7.99T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 0.782T + 61T^{2} \) |
| 67 | \( 1 + 9.90T + 67T^{2} \) |
| 71 | \( 1 - 0.364T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 7.28T + 79T^{2} \) |
| 83 | \( 1 + 5.69T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−7.54285266511861123821415387942, −6.89271954946884542715260479946, −6.34134649252764316390107116386, −5.82868702217114564778799895872, −4.85916467184187883876219994256, −4.28024019120717308242645832127, −3.43134124975411733336059568466, −1.47678075293104669451610853079, −1.30797450283784562812021453202, 0,
1.30797450283784562812021453202, 1.47678075293104669451610853079, 3.43134124975411733336059568466, 4.28024019120717308242645832127, 4.85916467184187883876219994256, 5.82868702217114564778799895872, 6.34134649252764316390107116386, 6.89271954946884542715260479946, 7.54285266511861123821415387942
