| L(s) = 1 | + 0.183·2-s − 1.96·4-s − 3.26·7-s − 0.726·8-s − 2·11-s − 0.296·13-s − 0.597·14-s + 3.79·16-s + 5.16·17-s + 1.73·19-s − 0.366·22-s − 0.879·23-s − 0.0542·26-s + 6.41·28-s − 5.91·29-s + 6.09·31-s + 2.14·32-s + 0.945·34-s + 8.08·37-s + 0.316·38-s + 1.01·41-s − 3.24·43-s + 3.93·44-s − 0.161·46-s − 4.21·47-s + 3.63·49-s + 0.582·52-s + ⋯ |
| L(s) = 1 | + 0.129·2-s − 0.983·4-s − 1.23·7-s − 0.256·8-s − 0.603·11-s − 0.0822·13-s − 0.159·14-s + 0.949·16-s + 1.25·17-s + 0.396·19-s − 0.0781·22-s − 0.183·23-s − 0.0106·26-s + 1.21·28-s − 1.09·29-s + 1.09·31-s + 0.379·32-s + 0.162·34-s + 1.32·37-s + 0.0514·38-s + 0.158·41-s − 0.494·43-s + 0.592·44-s − 0.0237·46-s − 0.615·47-s + 0.519·49-s + 0.0808·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.183T + 2T^{2} \) |
| 7 | \( 1 + 3.26T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 0.296T + 13T^{2} \) |
| 17 | \( 1 - 5.16T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 + 0.879T + 23T^{2} \) |
| 29 | \( 1 + 5.91T + 29T^{2} \) |
| 31 | \( 1 - 6.09T + 31T^{2} \) |
| 37 | \( 1 - 8.08T + 37T^{2} \) |
| 41 | \( 1 - 1.01T + 41T^{2} \) |
| 43 | \( 1 + 3.24T + 43T^{2} \) |
| 47 | \( 1 + 4.21T + 47T^{2} \) |
| 53 | \( 1 - 8.10T + 53T^{2} \) |
| 59 | \( 1 + 5.93T + 59T^{2} \) |
| 61 | \( 1 - 0.915T + 61T^{2} \) |
| 67 | \( 1 - 6.88T + 67T^{2} \) |
| 71 | \( 1 - 5.96T + 71T^{2} \) |
| 73 | \( 1 - 8.83T + 73T^{2} \) |
| 79 | \( 1 + 7.76T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 7.52T + 89T^{2} \) |
| 97 | \( 1 + 6.72T + 97T^{2} \) |
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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
−7.88139953065035955311370716634, −7.12303163074117168524601435733, −6.16521173509700735844454483058, −5.62436094566195638027783452724, −4.92568456538801024211696495420, −3.98439533358321117300645119423, −3.34187432488056250526484234457, −2.62802192399007126155552800489, −1.07364667626898514625881126734, 0,
1.07364667626898514625881126734, 2.62802192399007126155552800489, 3.34187432488056250526484234457, 3.98439533358321117300645119423, 4.92568456538801024211696495420, 5.62436094566195638027783452724, 6.16521173509700735844454483058, 7.12303163074117168524601435733, 7.88139953065035955311370716634
