| L(s) = 1 | + 2-s − 3-s + 4-s − 2.09·5-s − 6-s + 1.21·7-s + 8-s + 9-s − 2.09·10-s + 4.27·11-s − 12-s − 5.32·13-s + 1.21·14-s + 2.09·15-s + 16-s + 5.49·17-s + 18-s + 1.21·19-s − 2.09·20-s − 1.21·21-s + 4.27·22-s + 23-s − 24-s − 0.622·25-s − 5.32·26-s − 27-s + 1.21·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.935·5-s − 0.408·6-s + 0.458·7-s + 0.353·8-s + 0.333·9-s − 0.661·10-s + 1.28·11-s − 0.288·12-s − 1.47·13-s + 0.324·14-s + 0.540·15-s + 0.250·16-s + 1.33·17-s + 0.235·18-s + 0.278·19-s − 0.467·20-s − 0.264·21-s + 0.911·22-s + 0.208·23-s − 0.204·24-s − 0.124·25-s − 1.04·26-s − 0.192·27-s + 0.229·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.249748471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.249748471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 2.09T + 5T^{2} \) |
| 7 | \( 1 - 1.21T + 7T^{2} \) |
| 11 | \( 1 - 4.27T + 11T^{2} \) |
| 13 | \( 1 + 5.32T + 13T^{2} \) |
| 17 | \( 1 - 5.49T + 17T^{2} \) |
| 19 | \( 1 - 1.21T + 19T^{2} \) |
| 31 | \( 1 + 3.40T + 31T^{2} \) |
| 37 | \( 1 - 3.04T + 37T^{2} \) |
| 41 | \( 1 + 3.69T + 41T^{2} \) |
| 43 | \( 1 - 8.96T + 43T^{2} \) |
| 47 | \( 1 + 2.76T + 47T^{2} \) |
| 53 | \( 1 + 3.94T + 53T^{2} \) |
| 59 | \( 1 - 4.90T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 8.59T + 71T^{2} \) |
| 73 | \( 1 - 6.48T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 2.28T + 83T^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 - 3.02T + 97T^{2} \) |
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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
−8.096661497513299457604556452957, −7.60980957233041659800690628034, −6.97524134025214341338665766691, −6.19227710634806484709128334300, −5.27927676881286469109484497063, −4.73444797995678261224019819991, −3.93010751253390666015781198135, −3.26657626457563681950991518597, −1.98682124437432689105064265350, −0.813349962401600924061871787594,
0.813349962401600924061871787594, 1.98682124437432689105064265350, 3.26657626457563681950991518597, 3.93010751253390666015781198135, 4.73444797995678261224019819991, 5.27927676881286469109484497063, 6.19227710634806484709128334300, 6.97524134025214341338665766691, 7.60980957233041659800690628034, 8.096661497513299457604556452957
