L(s) = 1 | + 2.79·3-s − 3.17·5-s + 1.46·7-s + 4.79·9-s + 4.08·11-s + 4.91·13-s − 8.85·15-s + 3.89·17-s + 4.08·19-s + 4.09·21-s − 7.08·23-s + 5.05·25-s + 4.99·27-s − 1.48·29-s − 9.24·31-s + 11.3·33-s − 4.65·35-s + 2.10·37-s + 13.7·39-s − 6.48·41-s + 5.95·43-s − 15.1·45-s + 8.02·47-s − 4.84·49-s + 10.8·51-s + 4.69·53-s − 12.9·55-s + ⋯ |
L(s) = 1 | + 1.61·3-s − 1.41·5-s + 0.554·7-s + 1.59·9-s + 1.23·11-s + 1.36·13-s − 2.28·15-s + 0.944·17-s + 0.937·19-s + 0.894·21-s − 1.47·23-s + 1.01·25-s + 0.961·27-s − 0.275·29-s − 1.66·31-s + 1.98·33-s − 0.787·35-s + 0.346·37-s + 2.19·39-s − 1.01·41-s + 0.907·43-s − 2.26·45-s + 1.17·47-s − 0.692·49-s + 1.52·51-s + 0.644·53-s − 1.74·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.449902055\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.449902055\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 + 3.17T + 5T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 11 | \( 1 - 4.08T + 11T^{2} \) |
| 13 | \( 1 - 4.91T + 13T^{2} \) |
| 17 | \( 1 - 3.89T + 17T^{2} \) |
| 19 | \( 1 - 4.08T + 19T^{2} \) |
| 23 | \( 1 + 7.08T + 23T^{2} \) |
| 29 | \( 1 + 1.48T + 29T^{2} \) |
| 31 | \( 1 + 9.24T + 31T^{2} \) |
| 37 | \( 1 - 2.10T + 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 - 5.95T + 43T^{2} \) |
| 47 | \( 1 - 8.02T + 47T^{2} \) |
| 53 | \( 1 - 4.69T + 53T^{2} \) |
| 59 | \( 1 + 5.12T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 1.10T + 67T^{2} \) |
| 71 | \( 1 + 1.84T + 71T^{2} \) |
| 73 | \( 1 - 3.34T + 73T^{2} \) |
| 79 | \( 1 + 2.16T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 - 4.28T + 97T^{2} \) |
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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
−8.408652489691230921703202122132, −7.70101615844144170617375566252, −7.52322166657194417632312742488, −6.43748491656366111427431455342, −5.38098867675891907779061582759, −4.05348945794364889199936620426, −3.80960071729518161985854086023, −3.31333953403120909759337517554, −1.98330409474765685023284193907, −1.07799568295391165647600602240,
1.07799568295391165647600602240, 1.98330409474765685023284193907, 3.31333953403120909759337517554, 3.80960071729518161985854086023, 4.05348945794364889199936620426, 5.38098867675891907779061582759, 6.43748491656366111427431455342, 7.52322166657194417632312742488, 7.70101615844144170617375566252, 8.408652489691230921703202122132