| L(s) = 1 | − 1.04·2-s − 3-s − 0.904·4-s − 1.81·5-s + 1.04·6-s + 3.03·8-s + 9-s + 1.89·10-s − 0.445·11-s + 0.904·12-s + 2.67·13-s + 1.81·15-s − 1.37·16-s − 2.86·17-s − 1.04·18-s + 4.73·19-s + 1.64·20-s + 0.466·22-s + 3.59·23-s − 3.03·24-s − 1.71·25-s − 2.79·26-s − 27-s − 4.22·29-s − 1.89·30-s − 6.94·31-s − 4.64·32-s + ⋯ |
| L(s) = 1 | − 0.739·2-s − 0.577·3-s − 0.452·4-s − 0.810·5-s + 0.427·6-s + 1.07·8-s + 0.333·9-s + 0.600·10-s − 0.134·11-s + 0.261·12-s + 0.740·13-s + 0.468·15-s − 0.342·16-s − 0.695·17-s − 0.246·18-s + 1.08·19-s + 0.366·20-s + 0.0995·22-s + 0.750·23-s − 0.620·24-s − 0.342·25-s − 0.548·26-s − 0.192·27-s − 0.783·29-s − 0.346·30-s − 1.24·31-s − 0.821·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.04T + 2T^{2} \) |
| 5 | \( 1 + 1.81T + 5T^{2} \) |
| 11 | \( 1 + 0.445T + 11T^{2} \) |
| 13 | \( 1 - 2.67T + 13T^{2} \) |
| 17 | \( 1 + 2.86T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 - 3.59T + 23T^{2} \) |
| 29 | \( 1 + 4.22T + 29T^{2} \) |
| 31 | \( 1 + 6.94T + 31T^{2} \) |
| 37 | \( 1 + 9.17T + 37T^{2} \) |
| 43 | \( 1 + 7.79T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 - 7.26T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 5.02T + 79T^{2} \) |
| 83 | \( 1 - 7.60T + 83T^{2} \) |
| 89 | \( 1 + 2.20T + 89T^{2} \) |
| 97 | \( 1 + 1.75T + 97T^{2} \) |
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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
−7.51804981300612347959337982010, −7.41579493670292559595200522015, −6.44814276657809929450703303146, −5.38160891493742224354254136988, −4.99682779399232061383238797127, −3.89146690724364756804587083149, −3.56530305922292704762920250020, −1.99341598981168588209902825551, −0.956283294999502183027472815913, 0,
0.956283294999502183027472815913, 1.99341598981168588209902825551, 3.56530305922292704762920250020, 3.89146690724364756804587083149, 4.99682779399232061383238797127, 5.38160891493742224354254136988, 6.44814276657809929450703303146, 7.41579493670292559595200522015, 7.51804981300612347959337982010
