L(s) = 1 | + 2-s − 2.43·3-s + 4-s + 3.21·5-s − 2.43·6-s − 0.555·7-s + 8-s + 2.94·9-s + 3.21·10-s + 2.38·11-s − 2.43·12-s − 3.53·13-s − 0.555·14-s − 7.83·15-s + 16-s − 2.52·17-s + 2.94·18-s + 19-s + 3.21·20-s + 1.35·21-s + 2.38·22-s − 5.51·23-s − 2.43·24-s + 5.33·25-s − 3.53·26-s + 0.134·27-s − 0.555·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.40·3-s + 0.5·4-s + 1.43·5-s − 0.995·6-s − 0.209·7-s + 0.353·8-s + 0.981·9-s + 1.01·10-s + 0.718·11-s − 0.703·12-s − 0.980·13-s − 0.148·14-s − 2.02·15-s + 0.250·16-s − 0.611·17-s + 0.694·18-s + 0.229·19-s + 0.718·20-s + 0.295·21-s + 0.507·22-s − 1.15·23-s − 0.497·24-s + 1.06·25-s − 0.693·26-s + 0.0259·27-s − 0.104·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 + 2.43T + 3T^{2} \) |
| 5 | \( 1 - 3.21T + 5T^{2} \) |
| 7 | \( 1 + 0.555T + 7T^{2} \) |
| 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 + 3.53T + 13T^{2} \) |
| 17 | \( 1 + 2.52T + 17T^{2} \) |
| 23 | \( 1 + 5.51T + 23T^{2} \) |
| 29 | \( 1 + 4.31T + 29T^{2} \) |
| 31 | \( 1 - 3.40T + 31T^{2} \) |
| 37 | \( 1 - 3.58T + 37T^{2} \) |
| 41 | \( 1 + 0.595T + 41T^{2} \) |
| 43 | \( 1 + 6.33T + 43T^{2} \) |
| 47 | \( 1 + 4.42T + 47T^{2} \) |
| 53 | \( 1 + 8.35T + 53T^{2} \) |
| 59 | \( 1 - 2.18T + 59T^{2} \) |
| 61 | \( 1 + 3.16T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 5.55T + 71T^{2} \) |
| 73 | \( 1 + 1.24T + 73T^{2} \) |
| 79 | \( 1 + 1.74T + 79T^{2} \) |
| 83 | \( 1 + 3.53T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−7.04534398013378170767296336753, −6.37693806285539067634910282711, −6.16900940488384607081582092978, −5.43355482957026397870038265135, −4.88279700267367676299857191023, −4.23451209464222672633158853290, −3.08756198185492549946071403386, −2.12180698293286953151122935946, −1.41933230479102129309596849358, 0,
1.41933230479102129309596849358, 2.12180698293286953151122935946, 3.08756198185492549946071403386, 4.23451209464222672633158853290, 4.88279700267367676299857191023, 5.43355482957026397870038265135, 6.16900940488384607081582092978, 6.37693806285539067634910282711, 7.04534398013378170767296336753