| L(s) = 1 | + 2-s + 4-s + 1.11·5-s + 2.80·7-s + 8-s + 1.11·10-s + 4.77·11-s − 13-s + 2.80·14-s + 16-s − 4.69·17-s + 6.77·19-s + 1.11·20-s + 4.77·22-s + 2·23-s − 3.75·25-s − 26-s + 2.80·28-s + 0.229·29-s − 7.61·31-s + 32-s − 4.69·34-s + 3.12·35-s + 5.11·37-s + 6.77·38-s + 1.11·40-s − 6.77·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.498·5-s + 1.05·7-s + 0.353·8-s + 0.352·10-s + 1.44·11-s − 0.277·13-s + 0.748·14-s + 0.250·16-s − 1.13·17-s + 1.55·19-s + 0.249·20-s + 1.01·22-s + 0.417·23-s − 0.751·25-s − 0.196·26-s + 0.529·28-s + 0.0427·29-s − 1.36·31-s + 0.176·32-s − 0.805·34-s + 0.528·35-s + 0.840·37-s + 1.09·38-s + 0.176·40-s − 1.05·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.729903647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.729903647\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 1.11T + 5T^{2} \) |
| 7 | \( 1 - 2.80T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 17 | \( 1 + 4.69T + 17T^{2} \) |
| 19 | \( 1 - 6.77T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 0.229T + 29T^{2} \) |
| 31 | \( 1 + 7.61T + 31T^{2} \) |
| 37 | \( 1 - 5.11T + 37T^{2} \) |
| 41 | \( 1 + 6.77T + 41T^{2} \) |
| 43 | \( 1 - 8.69T + 43T^{2} \) |
| 47 | \( 1 + 7.47T + 47T^{2} \) |
| 53 | \( 1 - 1.77T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 3.83T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 - 0.165T + 79T^{2} \) |
| 83 | \( 1 + 2.39T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 7.00T + 97T^{2} \) |
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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
−9.272023449077188564258078065185, −8.257660183148223408026758440331, −7.38233819202284749655940112335, −6.69513300598838625323432997228, −5.82109775608287181696349898652, −5.05788252162865940597353541949, −4.30708495947186516356698618825, −3.40931177034335363036775863249, −2.14343749747332910483129209008, −1.33660084121077300124740369829,
1.33660084121077300124740369829, 2.14343749747332910483129209008, 3.40931177034335363036775863249, 4.30708495947186516356698618825, 5.05788252162865940597353541949, 5.82109775608287181696349898652, 6.69513300598838625323432997228, 7.38233819202284749655940112335, 8.257660183148223408026758440331, 9.272023449077188564258078065185
