L(s) = 1 | + 1.79·2-s + 1.23·4-s + 2.60·5-s + 1.37·7-s − 1.37·8-s + 4.69·10-s + 5.21·11-s − 0.609·13-s + 2.46·14-s − 4.94·16-s − 0.762·17-s + 5.43·19-s + 3.22·20-s + 9.38·22-s − 23-s + 1.79·25-s − 1.09·26-s + 1.69·28-s − 29-s + 3.96·31-s − 6.14·32-s − 1.37·34-s + 3.57·35-s − 3.31·37-s + 9.77·38-s − 3.57·40-s + 1.10·41-s + ⋯ |
L(s) = 1 | + 1.27·2-s + 0.618·4-s + 1.16·5-s + 0.517·7-s − 0.485·8-s + 1.48·10-s + 1.57·11-s − 0.169·13-s + 0.658·14-s − 1.23·16-s − 0.185·17-s + 1.24·19-s + 0.721·20-s + 2.00·22-s − 0.208·23-s + 0.359·25-s − 0.215·26-s + 0.320·28-s − 0.185·29-s + 0.712·31-s − 1.08·32-s − 0.235·34-s + 0.604·35-s − 0.544·37-s + 1.58·38-s − 0.565·40-s + 0.172·41-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
\(L(1)\) |
\(\approx\) |
\(5.624386909\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.624386909\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.79T + 2T^{2} \) |
| 5 | \( 1 - 2.60T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 - 5.21T + 11T^{2} \) |
| 13 | \( 1 + 0.609T + 13T^{2} \) |
| 17 | \( 1 + 0.762T + 17T^{2} \) |
| 19 | \( 1 - 5.43T + 19T^{2} \) |
| 31 | \( 1 - 3.96T + 31T^{2} \) |
| 37 | \( 1 + 3.31T + 37T^{2} \) |
| 41 | \( 1 - 1.10T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 4.97T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 2.20T + 71T^{2} \) |
| 73 | \( 1 - 3.58T + 73T^{2} \) |
| 79 | \( 1 + 7.27T + 79T^{2} \) |
| 83 | \( 1 - 3.93T + 83T^{2} \) |
| 89 | \( 1 - 1.67T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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
−8.006928969927613727752785598916, −6.99929191822099076604938089808, −6.45381093891233176389347569886, −5.73968867968021295534640380188, −5.30078203335679251076587815539, −4.43123955556116798054274042362, −3.84153613724456801363398713425, −2.91203876691062904076612220578, −2.04465523259547825795701381960, −1.12109256493675249475466339954,
1.12109256493675249475466339954, 2.04465523259547825795701381960, 2.91203876691062904076612220578, 3.84153613724456801363398713425, 4.43123955556116798054274042362, 5.30078203335679251076587815539, 5.73968867968021295534640380188, 6.45381093891233176389347569886, 6.99929191822099076604938089808, 8.006928969927613727752785598916