L(s) = 1 | − 1.96·2-s + 1.86·4-s − 2.31·5-s + 5.13·7-s + 0.268·8-s + 4.54·10-s + 2.78·11-s + 1.47·13-s − 10.1·14-s − 4.25·16-s + 1.99·17-s + 2.47·19-s − 4.31·20-s − 5.47·22-s + 23-s + 0.352·25-s − 2.89·26-s + 9.57·28-s − 29-s + 5.51·31-s + 7.82·32-s − 3.92·34-s − 11.8·35-s + 7.95·37-s − 4.85·38-s − 0.620·40-s − 2.86·41-s + ⋯ |
L(s) = 1 | − 1.38·2-s + 0.931·4-s − 1.03·5-s + 1.94·7-s + 0.0948·8-s + 1.43·10-s + 0.839·11-s + 0.408·13-s − 2.69·14-s − 1.06·16-s + 0.484·17-s + 0.567·19-s − 0.964·20-s − 1.16·22-s + 0.208·23-s + 0.0705·25-s − 0.567·26-s + 1.81·28-s − 0.185·29-s + 0.990·31-s + 1.38·32-s − 0.672·34-s − 2.00·35-s + 1.30·37-s − 0.788·38-s − 0.0981·40-s − 0.447·41-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
\(L(1)\) |
\(\approx\) |
\(1.223090417\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.223090417\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.96T + 2T^{2} \) |
| 5 | \( 1 + 2.31T + 5T^{2} \) |
| 7 | \( 1 - 5.13T + 7T^{2} \) |
| 11 | \( 1 - 2.78T + 11T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 - 1.99T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 31 | \( 1 - 5.51T + 31T^{2} \) |
| 37 | \( 1 - 7.95T + 37T^{2} \) |
| 41 | \( 1 + 2.86T + 41T^{2} \) |
| 43 | \( 1 - 6.65T + 43T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 - 1.25T + 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 - 7.80T + 61T^{2} \) |
| 67 | \( 1 + 4.27T + 67T^{2} \) |
| 71 | \( 1 - 3.71T + 71T^{2} \) |
| 73 | \( 1 + 2.30T + 73T^{2} \) |
| 79 | \( 1 - 9.24T + 79T^{2} \) |
| 83 | \( 1 - 1.97T + 83T^{2} \) |
| 89 | \( 1 - 4.99T + 89T^{2} \) |
| 97 | \( 1 + 2.35T + 97T^{2} \) |
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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
−8.107389200797217068592181518708, −7.66861369901240853041470089290, −7.17229405137029877783454709864, −6.13745309685545607993928101883, −5.08092236493279126971278576869, −4.40120798539928177902193290092, −3.77448005259922700904815169339, −2.41436001820516313211642507914, −1.37252798277468771351738588484, −0.849292558987414778674070010264,
0.849292558987414778674070010264, 1.37252798277468771351738588484, 2.41436001820516313211642507914, 3.77448005259922700904815169339, 4.40120798539928177902193290092, 5.08092236493279126971278576869, 6.13745309685545607993928101883, 7.17229405137029877783454709864, 7.66861369901240853041470089290, 8.107389200797217068592181518708