| L(s) = 1 | − 2-s + 2.06·3-s + 4-s + 3.73·5-s − 2.06·6-s + 4.82·7-s − 8-s + 1.25·9-s − 3.73·10-s + 2.07·11-s + 2.06·12-s − 13-s − 4.82·14-s + 7.70·15-s + 16-s + 4.11·17-s − 1.25·18-s − 1.79·19-s + 3.73·20-s + 9.95·21-s − 2.07·22-s − 7.82·23-s − 2.06·24-s + 8.93·25-s + 26-s − 3.59·27-s + 4.82·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.19·3-s + 0.5·4-s + 1.66·5-s − 0.842·6-s + 1.82·7-s − 0.353·8-s + 0.418·9-s − 1.18·10-s + 0.626·11-s + 0.595·12-s − 0.277·13-s − 1.28·14-s + 1.98·15-s + 0.250·16-s + 0.998·17-s − 0.296·18-s − 0.411·19-s + 0.834·20-s + 2.17·21-s − 0.442·22-s − 1.63·23-s − 0.421·24-s + 1.78·25-s + 0.196·26-s − 0.692·27-s + 0.912·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2678 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.411762860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.411762860\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 3 | \( 1 - 2.06T + 3T^{2} \) |
| 5 | \( 1 - 3.73T + 5T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 - 2.07T + 11T^{2} \) |
| 17 | \( 1 - 4.11T + 17T^{2} \) |
| 19 | \( 1 + 1.79T + 19T^{2} \) |
| 23 | \( 1 + 7.82T + 23T^{2} \) |
| 29 | \( 1 - 1.67T + 29T^{2} \) |
| 31 | \( 1 - 2.25T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 4.01T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 0.0980T + 61T^{2} \) |
| 67 | \( 1 - 1.67T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 3.87T + 73T^{2} \) |
| 79 | \( 1 - 4.29T + 79T^{2} \) |
| 83 | \( 1 - 0.687T + 83T^{2} \) |
| 89 | \( 1 - 5.70T + 89T^{2} \) |
| 97 | \( 1 - 5.85T + 97T^{2} \) |
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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
−8.772438943222710717018421224338, −8.304872413534628179285775041367, −7.65857089432741775510794025476, −6.71680678978064207526910488124, −5.74098855017048726915143918777, −5.11070775354756852186416283806, −3.91576649223837750493708967161, −2.70873312261039878210318259385, −1.82748262682609777445240196420, −1.54957680569094548464538120045,
1.54957680569094548464538120045, 1.82748262682609777445240196420, 2.70873312261039878210318259385, 3.91576649223837750493708967161, 5.11070775354756852186416283806, 5.74098855017048726915143918777, 6.71680678978064207526910488124, 7.65857089432741775510794025476, 8.304872413534628179285775041367, 8.772438943222710717018421224338
