| L(s) = 1 | − 2.40·2-s + 3.77·4-s − 1.03·7-s − 4.26·8-s − 4.74·11-s + 5.35·13-s + 2.48·14-s + 2.69·16-s + 1.99·17-s + 3.89·19-s + 11.4·22-s + 1.23·23-s − 12.8·26-s − 3.90·28-s − 10.1·29-s − 31-s + 2.05·32-s − 4.80·34-s + 5.71·37-s − 9.36·38-s + 2.69·41-s − 7.63·43-s − 17.9·44-s − 2.97·46-s + 6.19·47-s − 5.92·49-s + 20.2·52-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.88·4-s − 0.391·7-s − 1.50·8-s − 1.43·11-s + 1.48·13-s + 0.664·14-s + 0.673·16-s + 0.484·17-s + 0.893·19-s + 2.43·22-s + 0.258·23-s − 2.52·26-s − 0.738·28-s − 1.88·29-s − 0.179·31-s + 0.362·32-s − 0.823·34-s + 0.939·37-s − 1.51·38-s + 0.421·41-s − 1.16·43-s − 2.69·44-s − 0.438·46-s + 0.903·47-s − 0.846·49-s + 2.80·52-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.40T + 2T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 - 1.99T + 17T^{2} \) |
| 19 | \( 1 - 3.89T + 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 37 | \( 1 - 5.71T + 37T^{2} \) |
| 41 | \( 1 - 2.69T + 41T^{2} \) |
| 43 | \( 1 + 7.63T + 43T^{2} \) |
| 47 | \( 1 - 6.19T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 5.21T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 + 1.84T + 73T^{2} \) |
| 79 | \( 1 - 4.80T + 79T^{2} \) |
| 83 | \( 1 - 17.9T + 83T^{2} \) |
| 89 | \( 1 - 2.13T + 89T^{2} \) |
| 97 | \( 1 + 2.53T + 97T^{2} \) |
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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
−7.70334330035226561494974938953, −7.34578584586361405984315194528, −6.32167402190689470025664482926, −5.80580114639767893061010184315, −4.92108395203366247828417641003, −3.60933228415888759460600387293, −2.95724651431539271431666965851, −1.94949518273942346342492098869, −1.06510144487626898838196053229, 0,
1.06510144487626898838196053229, 1.94949518273942346342492098869, 2.95724651431539271431666965851, 3.60933228415888759460600387293, 4.92108395203366247828417641003, 5.80580114639767893061010184315, 6.32167402190689470025664482926, 7.34578584586361405984315194528, 7.70334330035226561494974938953
