| L(s) = 1 | + 3-s − 0.619·5-s + 0.954·7-s + 9-s + 6.25·11-s + 2.92·13-s − 0.619·15-s + 5.33·17-s + 2.94·19-s + 0.954·21-s − 2.33·23-s − 4.61·25-s + 27-s + 4.47·29-s − 3.45·31-s + 6.25·33-s − 0.590·35-s + 4.26·37-s + 2.92·39-s − 10.8·41-s − 1.32·43-s − 0.619·45-s + 3.37·47-s − 6.08·49-s + 5.33·51-s + 5.72·53-s − 3.87·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.276·5-s + 0.360·7-s + 0.333·9-s + 1.88·11-s + 0.811·13-s − 0.159·15-s + 1.29·17-s + 0.675·19-s + 0.208·21-s − 0.487·23-s − 0.923·25-s + 0.192·27-s + 0.830·29-s − 0.620·31-s + 1.08·33-s − 0.0998·35-s + 0.701·37-s + 0.468·39-s − 1.68·41-s − 0.201·43-s − 0.0923·45-s + 0.491·47-s − 0.869·49-s + 0.747·51-s + 0.786·53-s − 0.522·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.017242044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.017242044\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 0.619T + 5T^{2} \) |
| 7 | \( 1 - 0.954T + 7T^{2} \) |
| 11 | \( 1 - 6.25T + 11T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 - 5.33T + 17T^{2} \) |
| 19 | \( 1 - 2.94T + 19T^{2} \) |
| 23 | \( 1 + 2.33T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 3.45T + 31T^{2} \) |
| 37 | \( 1 - 4.26T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 1.32T + 43T^{2} \) |
| 47 | \( 1 - 3.37T + 47T^{2} \) |
| 53 | \( 1 - 5.72T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 0.604T + 67T^{2} \) |
| 71 | \( 1 + 9.74T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 6.74T + 79T^{2} \) |
| 83 | \( 1 - 0.00272T + 83T^{2} \) |
| 89 | \( 1 - 9.85T + 89T^{2} \) |
| 97 | \( 1 + 8.92T + 97T^{2} \) |
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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
−8.418287063498549629391656703945, −7.83624951897204112652879992861, −7.05687353359578761993534483911, −6.29256462131453708789440495676, −5.54673206307631952766946580353, −4.44840330873279896252549398171, −3.73183144566666042558635631024, −3.22140961854196545189341851937, −1.78361052107674969049509889923, −1.09130026746448953081455861403,
1.09130026746448953081455861403, 1.78361052107674969049509889923, 3.22140961854196545189341851937, 3.73183144566666042558635631024, 4.44840330873279896252549398171, 5.54673206307631952766946580353, 6.29256462131453708789440495676, 7.05687353359578761993534483911, 7.83624951897204112652879992861, 8.418287063498549629391656703945
