| L(s) = 1 | − 0.496·2-s − 1.75·4-s + 1.76·7-s + 1.86·8-s + 3.54·11-s + 4.37·13-s − 0.878·14-s + 2.57·16-s − 5.52·17-s + 6.04·19-s − 1.76·22-s + 6.47·23-s − 2.17·26-s − 3.10·28-s − 2.89·29-s − 31-s − 5.01·32-s + 2.74·34-s + 10.0·37-s − 3.00·38-s + 10.8·41-s + 2.90·43-s − 6.21·44-s − 3.21·46-s + 7.89·47-s − 3.87·49-s − 7.66·52-s + ⋯ |
| L(s) = 1 | − 0.351·2-s − 0.876·4-s + 0.668·7-s + 0.659·8-s + 1.06·11-s + 1.21·13-s − 0.234·14-s + 0.644·16-s − 1.33·17-s + 1.38·19-s − 0.375·22-s + 1.34·23-s − 0.426·26-s − 0.585·28-s − 0.537·29-s − 0.179·31-s − 0.885·32-s + 0.470·34-s + 1.65·37-s − 0.487·38-s + 1.70·41-s + 0.443·43-s − 0.937·44-s − 0.474·46-s + 1.15·47-s − 0.553·49-s − 1.06·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.907788999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.907788999\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.496T + 2T^{2} \) |
| 7 | \( 1 - 1.76T + 7T^{2} \) |
| 11 | \( 1 - 3.54T + 11T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 + 5.52T + 17T^{2} \) |
| 19 | \( 1 - 6.04T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 - 7.89T + 47T^{2} \) |
| 53 | \( 1 + 3.26T + 53T^{2} \) |
| 59 | \( 1 - 9.66T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 2.72T + 71T^{2} \) |
| 73 | \( 1 + 7.66T + 73T^{2} \) |
| 79 | \( 1 - 3.25T + 79T^{2} \) |
| 83 | \( 1 + 5.94T + 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 - 1.46T + 97T^{2} \) |
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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
−8.021683673447998848937476201014, −7.41645990794563479313970387571, −6.60076441904854652154141508258, −5.79265906769458219858994345782, −5.07433870358701211902520988656, −4.23441167671111454233066529666, −3.85526535391385935994288284908, −2.72014903557932771452266227203, −1.41228671943985273385379714601, −0.876790611986400371339160622383,
0.876790611986400371339160622383, 1.41228671943985273385379714601, 2.72014903557932771452266227203, 3.85526535391385935994288284908, 4.23441167671111454233066529666, 5.07433870358701211902520988656, 5.79265906769458219858994345782, 6.60076441904854652154141508258, 7.41645990794563479313970387571, 8.021683673447998848937476201014
