L(s) = 1 | + 3-s + 5-s − 3.83·7-s + 9-s + 1.14·11-s − 3.53·13-s + 15-s − 6.97·17-s + 19-s − 3.83·21-s + 8.97·23-s + 25-s + 27-s + 0.853·29-s + 4.39·31-s + 1.14·33-s − 3.83·35-s + 1.83·37-s − 3.53·39-s − 8.51·41-s + 10.1·43-s + 45-s − 0.978·47-s + 7.68·49-s − 6.97·51-s + 6.97·53-s + 1.14·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.44·7-s + 0.333·9-s + 0.345·11-s − 0.981·13-s + 0.258·15-s − 1.69·17-s + 0.229·19-s − 0.836·21-s + 1.87·23-s + 0.200·25-s + 0.192·27-s + 0.158·29-s + 0.789·31-s + 0.199·33-s − 0.647·35-s + 0.301·37-s − 0.566·39-s − 1.33·41-s + 1.54·43-s + 0.149·45-s − 0.142·47-s + 1.09·49-s − 0.977·51-s + 0.958·53-s + 0.154·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.969877667\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.969877667\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 + 3.53T + 13T^{2} \) |
| 17 | \( 1 + 6.97T + 17T^{2} \) |
| 23 | \( 1 - 8.97T + 23T^{2} \) |
| 29 | \( 1 - 0.853T + 29T^{2} \) |
| 31 | \( 1 - 4.39T + 31T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 + 8.51T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 0.978T + 47T^{2} \) |
| 53 | \( 1 - 6.97T + 53T^{2} \) |
| 59 | \( 1 - 6.29T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 0.585T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 7.37T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 7.93T + 89T^{2} \) |
| 97 | \( 1 - 1.24T + 97T^{2} \) |
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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
−8.623363505951479634316731172432, −7.41924121037263209694401441584, −6.77621103687427947642378131460, −6.46947821770975578865892590576, −5.33946185409008379901174344745, −4.56447920738468185414852580008, −3.64451772136082626038309899625, −2.78236509784746885664175478420, −2.25073185862266782942922809410, −0.73685881756563129953925303130,
0.73685881756563129953925303130, 2.25073185862266782942922809410, 2.78236509784746885664175478420, 3.64451772136082626038309899625, 4.56447920738468185414852580008, 5.33946185409008379901174344745, 6.46947821770975578865892590576, 6.77621103687427947642378131460, 7.41924121037263209694401441584, 8.623363505951479634316731172432