| L(s) = 1 | + 1.21·2-s − 0.534·4-s − 5-s − 3.06·8-s − 1.21·10-s + 0.255·11-s + 0.744·13-s − 2.64·16-s + 1.21·17-s − 1.11·19-s + 0.534·20-s + 0.308·22-s + 7.85·23-s + 25-s + 0.901·26-s + 3.32·29-s − 6.91·31-s + 2.93·32-s + 1.46·34-s + 9.27·37-s − 1.34·38-s + 3.06·40-s − 8.81·41-s + 5.70·43-s − 0.136·44-s + 9.51·46-s + 7.91·47-s + ⋯ |
| L(s) = 1 | + 0.856·2-s − 0.267·4-s − 0.447·5-s − 1.08·8-s − 0.382·10-s + 0.0769·11-s + 0.206·13-s − 0.661·16-s + 0.293·17-s − 0.255·19-s + 0.119·20-s + 0.0658·22-s + 1.63·23-s + 0.200·25-s + 0.176·26-s + 0.617·29-s − 1.24·31-s + 0.518·32-s + 0.251·34-s + 1.52·37-s − 0.218·38-s + 0.485·40-s − 1.37·41-s + 0.869·43-s − 0.0205·44-s + 1.40·46-s + 1.15·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.033074808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.033074808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 11 | \( 1 - 0.255T + 11T^{2} \) |
| 13 | \( 1 - 0.744T + 13T^{2} \) |
| 17 | \( 1 - 1.21T + 17T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 23 | \( 1 - 7.85T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 + 6.91T + 31T^{2} \) |
| 37 | \( 1 - 9.27T + 37T^{2} \) |
| 41 | \( 1 + 8.81T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 - 7.91T + 47T^{2} \) |
| 53 | \( 1 - 2.42T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 9.16T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 0.435T + 73T^{2} \) |
| 79 | \( 1 - 2.37T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 8.44T + 97T^{2} \) |
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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
−8.956127132038498609115276083397, −8.400167624291976641407730655164, −7.37695671557945963759290273181, −6.62498979943563349241069514077, −5.67602459676973639793071221200, −5.02053116856084805829482934194, −4.16674521254184114629255014465, −3.47258209311682372464086372953, −2.54181851497783555226384241740, −0.828052157840321367741376217565,
0.828052157840321367741376217565, 2.54181851497783555226384241740, 3.47258209311682372464086372953, 4.16674521254184114629255014465, 5.02053116856084805829482934194, 5.67602459676973639793071221200, 6.62498979943563349241069514077, 7.37695671557945963759290273181, 8.400167624291976641407730655164, 8.956127132038498609115276083397
