| L(s) = 1 | + 1.89·2-s − 1.57·3-s + 1.60·4-s − 2.98·6-s − 0.327·7-s − 0.758·8-s − 0.528·9-s + 5.23·11-s − 2.51·12-s + 1.70·13-s − 0.621·14-s − 4.63·16-s + 4.80·17-s − 1.00·18-s + 6.79·19-s + 0.514·21-s + 9.93·22-s − 7.91·23-s + 1.19·24-s + 3.23·26-s + 5.54·27-s − 0.524·28-s + 7.79·29-s − 1.21·31-s − 7.28·32-s − 8.23·33-s + 9.12·34-s + ⋯ |
| L(s) = 1 | + 1.34·2-s − 0.907·3-s + 0.800·4-s − 1.21·6-s − 0.123·7-s − 0.268·8-s − 0.176·9-s + 1.57·11-s − 0.726·12-s + 0.473·13-s − 0.166·14-s − 1.15·16-s + 1.16·17-s − 0.236·18-s + 1.55·19-s + 0.112·21-s + 2.11·22-s − 1.65·23-s + 0.243·24-s + 0.634·26-s + 1.06·27-s − 0.0990·28-s + 1.44·29-s − 0.217·31-s − 1.28·32-s − 1.43·33-s + 1.56·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.461231256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.461231256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 1.89T + 2T^{2} \) |
| 3 | \( 1 + 1.57T + 3T^{2} \) |
| 7 | \( 1 + 0.327T + 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 - 1.70T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 - 6.79T + 19T^{2} \) |
| 23 | \( 1 + 7.91T + 23T^{2} \) |
| 29 | \( 1 - 7.79T + 29T^{2} \) |
| 31 | \( 1 + 1.21T + 31T^{2} \) |
| 37 | \( 1 - 3.08T + 37T^{2} \) |
| 41 | \( 1 - 7.96T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 53 | \( 1 - 3.23T + 53T^{2} \) |
| 59 | \( 1 + 9.68T + 59T^{2} \) |
| 61 | \( 1 + 3.54T + 61T^{2} \) |
| 67 | \( 1 - 9.31T + 67T^{2} \) |
| 71 | \( 1 + 6.17T + 71T^{2} \) |
| 73 | \( 1 + 1.38T + 73T^{2} \) |
| 79 | \( 1 + 4.45T + 79T^{2} \) |
| 83 | \( 1 - 1.72T + 83T^{2} \) |
| 89 | \( 1 + 4.13T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−9.817476649582825245718954166345, −9.123731855131781567423609429213, −7.959745782565772429553175603501, −6.82107717589052006400999171952, −5.94409782936820311537040627755, −5.75782541275938007117218825660, −4.59527397760634139543712260089, −3.82146597439224701388733698510, −2.90750633436738379445456290188, −1.08274244712567284864616615281,
1.08274244712567284864616615281, 2.90750633436738379445456290188, 3.82146597439224701388733698510, 4.59527397760634139543712260089, 5.75782541275938007117218825660, 5.94409782936820311537040627755, 6.82107717589052006400999171952, 7.959745782565772429553175603501, 9.123731855131781567423609429213, 9.817476649582825245718954166345
