| L(s) = 1 | + 2-s − 1.34·3-s + 4-s + 5-s − 1.34·6-s + 1.63·7-s + 8-s − 1.17·9-s + 10-s + 11-s − 1.34·12-s + 2.32·13-s + 1.63·14-s − 1.34·15-s + 16-s − 4.38·17-s − 1.17·18-s − 2.37·19-s + 20-s − 2.21·21-s + 22-s − 3.15·23-s − 1.34·24-s + 25-s + 2.32·26-s + 5.63·27-s + 1.63·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.779·3-s + 0.5·4-s + 0.447·5-s − 0.551·6-s + 0.619·7-s + 0.353·8-s − 0.392·9-s + 0.316·10-s + 0.301·11-s − 0.389·12-s + 0.645·13-s + 0.437·14-s − 0.348·15-s + 0.250·16-s − 1.06·17-s − 0.277·18-s − 0.543·19-s + 0.223·20-s − 0.482·21-s + 0.213·22-s − 0.657·23-s − 0.275·24-s + 0.200·25-s + 0.456·26-s + 1.08·27-s + 0.309·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 + 1.34T + 3T^{2} \) |
| 7 | \( 1 - 1.63T + 7T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 23 | \( 1 + 3.15T + 23T^{2} \) |
| 29 | \( 1 + 6.28T + 29T^{2} \) |
| 31 | \( 1 + 2.49T + 31T^{2} \) |
| 37 | \( 1 - 6.14T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 7.75T + 43T^{2} \) |
| 47 | \( 1 + 4.50T + 47T^{2} \) |
| 53 | \( 1 + 7.19T + 53T^{2} \) |
| 59 | \( 1 - 6.79T + 59T^{2} \) |
| 61 | \( 1 + 7.93T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 7.90T + 71T^{2} \) |
| 79 | \( 1 - 6.82T + 79T^{2} \) |
| 83 | \( 1 - 0.433T + 83T^{2} \) |
| 89 | \( 1 - 9.07T + 89T^{2} \) |
| 97 | \( 1 + 0.217T + 97T^{2} \) |
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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
−7.30350511128080656761028995090, −6.37091292257893880112237150541, −6.16968370203303746299636809141, −5.40822309115317864921166847667, −4.74751124770700659982149982995, −4.10970079015596617465620118241, −3.19162223194872213328052452196, −2.16563963920646624524996894778, −1.46146126364624604702189874517, 0,
1.46146126364624604702189874517, 2.16563963920646624524996894778, 3.19162223194872213328052452196, 4.10970079015596617465620118241, 4.74751124770700659982149982995, 5.40822309115317864921166847667, 6.16968370203303746299636809141, 6.37091292257893880112237150541, 7.30350511128080656761028995090
