L(s) = 1 | + 2.15·2-s − 2.96·3-s + 2.66·4-s − 3.06·5-s − 6.39·6-s − 7-s + 1.43·8-s + 5.76·9-s − 6.61·10-s − 5.60·11-s − 7.88·12-s − 2.28·13-s − 2.15·14-s + 9.06·15-s − 2.23·16-s + 4.43·17-s + 12.4·18-s + 4.31·19-s − 8.15·20-s + 2.96·21-s − 12.1·22-s − 2.19·23-s − 4.24·24-s + 4.37·25-s − 4.93·26-s − 8.18·27-s − 2.66·28-s + ⋯ |
L(s) = 1 | + 1.52·2-s − 1.70·3-s + 1.33·4-s − 1.36·5-s − 2.60·6-s − 0.377·7-s + 0.506·8-s + 1.92·9-s − 2.09·10-s − 1.68·11-s − 2.27·12-s − 0.633·13-s − 0.577·14-s + 2.34·15-s − 0.558·16-s + 1.07·17-s + 2.93·18-s + 0.989·19-s − 1.82·20-s + 0.646·21-s − 2.58·22-s − 0.457·23-s − 0.865·24-s + 0.874·25-s − 0.967·26-s − 1.57·27-s − 0.503·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 3 | \( 1 + 2.96T + 3T^{2} \) |
| 5 | \( 1 + 3.06T + 5T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 + 2.28T + 13T^{2} \) |
| 17 | \( 1 - 4.43T + 17T^{2} \) |
| 19 | \( 1 - 4.31T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 + 0.786T + 29T^{2} \) |
| 31 | \( 1 + 6.29T + 31T^{2} \) |
| 37 | \( 1 - 7.79T + 37T^{2} \) |
| 41 | \( 1 + 8.60T + 41T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 4.68T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 4.03T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 2.91T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 5.08T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−11.56883776898643396407541914727, −10.90727952935556983458233678252, −9.898959634421845427043315668041, −7.75949977326870638282891257088, −7.11897423984301049894154199287, −5.77471753519870333621096111533, −5.23892423445256605001444590182, −4.31899926870652576622389521849, −3.12760597957516356827974649862, 0,
3.12760597957516356827974649862, 4.31899926870652576622389521849, 5.23892423445256605001444590182, 5.77471753519870333621096111533, 7.11897423984301049894154199287, 7.75949977326870638282891257088, 9.898959634421845427043315668041, 10.90727952935556983458233678252, 11.56883776898643396407541914727