| L(s) = 1 | + 2-s − 1.31·3-s + 4-s − 5-s − 1.31·6-s − 3.01·7-s + 8-s − 1.26·9-s − 10-s − 4.14·11-s − 1.31·12-s + 4.55·13-s − 3.01·14-s + 1.31·15-s + 16-s − 1.18·17-s − 1.26·18-s + 2.76·19-s − 20-s + 3.96·21-s − 4.14·22-s − 3.56·23-s − 1.31·24-s + 25-s + 4.55·26-s + 5.61·27-s − 3.01·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.760·3-s + 0.5·4-s − 0.447·5-s − 0.537·6-s − 1.13·7-s + 0.353·8-s − 0.421·9-s − 0.316·10-s − 1.25·11-s − 0.380·12-s + 1.26·13-s − 0.805·14-s + 0.340·15-s + 0.250·16-s − 0.287·17-s − 0.298·18-s + 0.635·19-s − 0.223·20-s + 0.866·21-s − 0.884·22-s − 0.744·23-s − 0.268·24-s + 0.200·25-s + 0.893·26-s + 1.08·27-s − 0.569·28-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
| \(L(1)\) |
\(\approx\) |
\(1.134844079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.134844079\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.31T + 3T^{2} \) |
| 7 | \( 1 + 3.01T + 7T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 - 4.55T + 13T^{2} \) |
| 17 | \( 1 + 1.18T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 + 3.56T + 23T^{2} \) |
| 29 | \( 1 + 6.21T + 29T^{2} \) |
| 31 | \( 1 - 2.36T + 31T^{2} \) |
| 37 | \( 1 + 9.95T + 37T^{2} \) |
| 41 | \( 1 - 3.57T + 41T^{2} \) |
| 43 | \( 1 + 7.80T + 43T^{2} \) |
| 47 | \( 1 - 4.65T + 47T^{2} \) |
| 53 | \( 1 + 1.53T + 53T^{2} \) |
| 59 | \( 1 + 7.45T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 5.86T + 71T^{2} \) |
| 73 | \( 1 - 9.19T + 73T^{2} \) |
| 79 | \( 1 - 5.83T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 3.67T + 97T^{2} \) |
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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
−8.326556930038366458230004316143, −7.59290817121477373751822302459, −6.69035745299732010506731185344, −6.15026485945259872629161418125, −5.49023231596201030983871553742, −4.87904573831942042200597213882, −3.64412328578329311940497055923, −3.30583027973472255274808347471, −2.17502214587700882066375105602, −0.53990308341126136338677774621,
0.53990308341126136338677774621, 2.17502214587700882066375105602, 3.30583027973472255274808347471, 3.64412328578329311940497055923, 4.87904573831942042200597213882, 5.49023231596201030983871553742, 6.15026485945259872629161418125, 6.69035745299732010506731185344, 7.59290817121477373751822302459, 8.326556930038366458230004316143
