| L(s) = 1 | + 1.61·2-s + 0.618·4-s + 5-s − 3·7-s − 2.23·8-s + 1.61·10-s − 3.47·11-s − 1.76·13-s − 4.85·14-s − 4.85·16-s + 5.47·17-s − 5.70·19-s + 0.618·20-s − 5.61·22-s + 25-s − 2.85·26-s − 1.85·28-s − 29-s − 8·31-s − 3.38·32-s + 8.85·34-s − 3·35-s − 8·37-s − 9.23·38-s − 2.23·40-s − 4.47·41-s − 1.23·43-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.309·4-s + 0.447·5-s − 1.13·7-s − 0.790·8-s + 0.511·10-s − 1.04·11-s − 0.489·13-s − 1.29·14-s − 1.21·16-s + 1.32·17-s − 1.30·19-s + 0.138·20-s − 1.19·22-s + 0.200·25-s − 0.559·26-s − 0.350·28-s − 0.185·29-s − 1.43·31-s − 0.597·32-s + 1.51·34-s − 0.507·35-s − 1.31·37-s − 1.49·38-s − 0.353·40-s − 0.698·41-s − 0.188·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 - 5.47T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 - 6.70T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 0.763T + 59T^{2} \) |
| 61 | \( 1 - 7.70T + 61T^{2} \) |
| 67 | \( 1 + 2.52T + 67T^{2} \) |
| 71 | \( 1 + 2.76T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 9.70T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 7.23T + 97T^{2} \) |
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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
−9.328163951011667657685019483990, −8.517157331709636875718521010631, −7.33805759860330208546512951483, −6.52717907393875981455482053830, −5.61319524016706448913866919633, −5.20504930187934825297667122058, −3.96029884260651070620463162534, −3.18858244896882346992177232948, −2.26582325386719369073341016964, 0,
2.26582325386719369073341016964, 3.18858244896882346992177232948, 3.96029884260651070620463162534, 5.20504930187934825297667122058, 5.61319524016706448913866919633, 6.52717907393875981455482053830, 7.33805759860330208546512951483, 8.517157331709636875718521010631, 9.328163951011667657685019483990
