| L(s) = 1 | + 0.364·2-s + 3-s − 1.86·4-s + 0.364·6-s − 2.24·7-s − 1.40·8-s + 9-s + 4.63·11-s − 1.86·12-s − 3.75·13-s − 0.818·14-s + 3.22·16-s + 5.36·17-s + 0.364·18-s − 5.66·19-s − 2.24·21-s + 1.68·22-s − 1.32·23-s − 1.40·24-s − 1.36·26-s + 27-s + 4.20·28-s + 8.86·29-s + 1.21·31-s + 3.98·32-s + 4.63·33-s + 1.95·34-s + ⋯ |
| L(s) = 1 | + 0.257·2-s + 0.577·3-s − 0.933·4-s + 0.148·6-s − 0.850·7-s − 0.497·8-s + 0.333·9-s + 1.39·11-s − 0.539·12-s − 1.04·13-s − 0.218·14-s + 0.805·16-s + 1.30·17-s + 0.0858·18-s − 1.29·19-s − 0.490·21-s + 0.360·22-s − 0.275·23-s − 0.287·24-s − 0.268·26-s + 0.192·27-s + 0.793·28-s + 1.64·29-s + 0.217·31-s + 0.705·32-s + 0.807·33-s + 0.335·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.758040543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.758040543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.364T + 2T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 - 4.63T + 11T^{2} \) |
| 13 | \( 1 + 3.75T + 13T^{2} \) |
| 17 | \( 1 - 5.36T + 17T^{2} \) |
| 19 | \( 1 + 5.66T + 19T^{2} \) |
| 23 | \( 1 + 1.32T + 23T^{2} \) |
| 29 | \( 1 - 8.86T + 29T^{2} \) |
| 31 | \( 1 - 1.21T + 31T^{2} \) |
| 37 | \( 1 - 2.15T + 37T^{2} \) |
| 41 | \( 1 - 3.49T + 41T^{2} \) |
| 43 | \( 1 + 1.16T + 43T^{2} \) |
| 47 | \( 1 - 2.36T + 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 - 0.367T + 59T^{2} \) |
| 61 | \( 1 - 5.29T + 61T^{2} \) |
| 67 | \( 1 + 8.06T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−9.227133257356454911144860862072, −8.587875716883097288935578984918, −7.80145170805300305240330973260, −6.74863606740909030532317697292, −6.11477046901594172325818644753, −5.00183291225306081581359057523, −4.12942932164495422157477021775, −3.51141197785353861299989016366, −2.47512843354189512259585193224, −0.858031163400899411759154944994,
0.858031163400899411759154944994, 2.47512843354189512259585193224, 3.51141197785353861299989016366, 4.12942932164495422157477021775, 5.00183291225306081581359057523, 6.11477046901594172325818644753, 6.74863606740909030532317697292, 7.80145170805300305240330973260, 8.587875716883097288935578984918, 9.227133257356454911144860862072
