| L(s) = 1 | + 1.31·2-s − 0.260·4-s − 2.51·5-s − 2.25·7-s − 2.98·8-s − 3.31·10-s + 0.206·11-s − 0.720·13-s − 2.97·14-s − 3.41·16-s − 2.26·17-s − 4.58·19-s + 0.655·20-s + 0.272·22-s − 23-s + 1.31·25-s − 0.950·26-s + 0.587·28-s − 29-s − 0.615·31-s + 1.46·32-s − 2.98·34-s + 5.66·35-s + 1.84·37-s − 6.04·38-s + 7.49·40-s + 3.39·41-s + ⋯ |
| L(s) = 1 | + 0.932·2-s − 0.130·4-s − 1.12·5-s − 0.851·7-s − 1.05·8-s − 1.04·10-s + 0.0623·11-s − 0.199·13-s − 0.793·14-s − 0.852·16-s − 0.548·17-s − 1.05·19-s + 0.146·20-s + 0.0581·22-s − 0.208·23-s + 0.263·25-s − 0.186·26-s + 0.111·28-s − 0.185·29-s − 0.110·31-s + 0.259·32-s − 0.511·34-s + 0.957·35-s + 0.302·37-s − 0.981·38-s + 1.18·40-s + 0.530·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7182787633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7182787633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.31T + 2T^{2} \) |
| 5 | \( 1 + 2.51T + 5T^{2} \) |
| 7 | \( 1 + 2.25T + 7T^{2} \) |
| 11 | \( 1 - 0.206T + 11T^{2} \) |
| 13 | \( 1 + 0.720T + 13T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 19 | \( 1 + 4.58T + 19T^{2} \) |
| 31 | \( 1 + 0.615T + 31T^{2} \) |
| 37 | \( 1 - 1.84T + 37T^{2} \) |
| 41 | \( 1 - 3.39T + 41T^{2} \) |
| 43 | \( 1 + 8.50T + 43T^{2} \) |
| 47 | \( 1 + 5.89T + 47T^{2} \) |
| 53 | \( 1 + 5.61T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 - 9.37T + 61T^{2} \) |
| 67 | \( 1 + 9.43T + 67T^{2} \) |
| 71 | \( 1 + 1.35T + 71T^{2} \) |
| 73 | \( 1 + 1.27T + 73T^{2} \) |
| 79 | \( 1 - 9.36T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 3.61T + 89T^{2} \) |
| 97 | \( 1 - 1.72T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.115141209563644433575143272765, −7.24634673500471507551706097470, −6.45869039993390428963177139875, −6.02555646395716858223085438025, −4.92213024907819756867589143723, −4.43744899661310374654275864577, −3.66285829373687060448852572224, −3.22124816691401522485198495773, −2.15438108284872180995596219527, −0.36345353685894041128122060480,
0.36345353685894041128122060480, 2.15438108284872180995596219527, 3.22124816691401522485198495773, 3.66285829373687060448852572224, 4.43744899661310374654275864577, 4.92213024907819756867589143723, 6.02555646395716858223085438025, 6.45869039993390428963177139875, 7.24634673500471507551706097470, 8.115141209563644433575143272765
