L(s) = 1 | + 2-s + 1.82·3-s + 4-s − 5-s + 1.82·6-s − 2.93·7-s + 8-s + 0.327·9-s − 10-s + 3.64·11-s + 1.82·12-s − 5.85·13-s − 2.93·14-s − 1.82·15-s + 16-s − 2.30·17-s + 0.327·18-s + 3.16·19-s − 20-s − 5.36·21-s + 3.64·22-s − 3.01·23-s + 1.82·24-s + 25-s − 5.85·26-s − 4.87·27-s − 2.93·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.05·3-s + 0.5·4-s − 0.447·5-s + 0.744·6-s − 1.11·7-s + 0.353·8-s + 0.109·9-s − 0.316·10-s + 1.10·11-s + 0.526·12-s − 1.62·13-s − 0.785·14-s − 0.471·15-s + 0.250·16-s − 0.559·17-s + 0.0772·18-s + 0.725·19-s − 0.223·20-s − 1.17·21-s + 0.777·22-s − 0.628·23-s + 0.372·24-s + 0.200·25-s − 1.14·26-s − 0.938·27-s − 0.555·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 1.82T + 3T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 - 3.64T + 11T^{2} \) |
| 13 | \( 1 + 5.85T + 13T^{2} \) |
| 17 | \( 1 + 2.30T + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 + 3.01T + 23T^{2} \) |
| 29 | \( 1 + 5.30T + 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 + 1.21T + 37T^{2} \) |
| 41 | \( 1 - 0.519T + 41T^{2} \) |
| 43 | \( 1 + 4.28T + 43T^{2} \) |
| 47 | \( 1 - 0.0999T + 47T^{2} \) |
| 53 | \( 1 + 6.44T + 53T^{2} \) |
| 59 | \( 1 - 6.79T + 59T^{2} \) |
| 61 | \( 1 + 0.500T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 6.59T + 71T^{2} \) |
| 73 | \( 1 - 4.75T + 73T^{2} \) |
| 79 | \( 1 + 1.23T + 79T^{2} \) |
| 83 | \( 1 + 9.00T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 2.07T + 97T^{2} \) |
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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
−7.957556057415020885341196604718, −7.25406794876710819414600482618, −6.77334687646313982405391211309, −5.84265896526222630464993667501, −4.96976330617044517952170141360, −3.92395804863325476934395545383, −3.51364868987094787418646488101, −2.70518396063675799109818462275, −1.87759096507658055105667345235, 0,
1.87759096507658055105667345235, 2.70518396063675799109818462275, 3.51364868987094787418646488101, 3.92395804863325476934395545383, 4.96976330617044517952170141360, 5.84265896526222630464993667501, 6.77334687646313982405391211309, 7.25406794876710819414600482618, 7.957556057415020885341196604718