| L(s) = 1 | + 1.05·2-s − 3-s − 0.880·4-s − 0.912·5-s − 1.05·6-s − 3.63·7-s − 3.04·8-s + 9-s − 0.965·10-s − 1.15·11-s + 0.880·12-s − 13-s − 3.84·14-s + 0.912·15-s − 1.46·16-s − 2.09·17-s + 1.05·18-s − 0.646·19-s + 0.803·20-s + 3.63·21-s − 1.21·22-s − 8.89·23-s + 3.04·24-s − 4.16·25-s − 1.05·26-s − 27-s + 3.19·28-s + ⋯ |
| L(s) = 1 | + 0.748·2-s − 0.577·3-s − 0.440·4-s − 0.408·5-s − 0.432·6-s − 1.37·7-s − 1.07·8-s + 0.333·9-s − 0.305·10-s − 0.347·11-s + 0.254·12-s − 0.277·13-s − 1.02·14-s + 0.235·15-s − 0.366·16-s − 0.507·17-s + 0.249·18-s − 0.148·19-s + 0.179·20-s + 0.792·21-s − 0.259·22-s − 1.85·23-s + 0.622·24-s − 0.833·25-s − 0.207·26-s − 0.192·27-s + 0.604·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3548301314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3548301314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 1.05T + 2T^{2} \) |
| 5 | \( 1 + 0.912T + 5T^{2} \) |
| 7 | \( 1 + 3.63T + 7T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 17 | \( 1 + 2.09T + 17T^{2} \) |
| 19 | \( 1 + 0.646T + 19T^{2} \) |
| 23 | \( 1 + 8.89T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 + 4.66T + 31T^{2} \) |
| 37 | \( 1 + 1.70T + 37T^{2} \) |
| 41 | \( 1 + 8.06T + 41T^{2} \) |
| 43 | \( 1 - 0.912T + 43T^{2} \) |
| 47 | \( 1 - 4.88T + 47T^{2} \) |
| 53 | \( 1 + 3.15T + 53T^{2} \) |
| 59 | \( 1 - 1.05T + 59T^{2} \) |
| 61 | \( 1 + 4.06T + 61T^{2} \) |
| 67 | \( 1 - 3.87T + 67T^{2} \) |
| 71 | \( 1 - 2.77T + 71T^{2} \) |
| 73 | \( 1 + 1.56T + 73T^{2} \) |
| 79 | \( 1 - 1.26T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 4.91T + 97T^{2} \) |
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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
−8.445582876599294828874690717201, −7.62009927676345503482430811411, −6.67501556111838635781705065360, −6.12990959550144331025575027160, −5.50163165871625387686679697970, −4.61686840276251157558655438384, −3.89316085858156176184063168892, −3.30875622415090114830442369828, −2.19345365296022509557720848646, −0.29334323726556330932125815916,
0.29334323726556330932125815916, 2.19345365296022509557720848646, 3.30875622415090114830442369828, 3.89316085858156176184063168892, 4.61686840276251157558655438384, 5.50163165871625387686679697970, 6.12990959550144331025575027160, 6.67501556111838635781705065360, 7.62009927676345503482430811411, 8.445582876599294828874690717201
