| L(s) = 1 | − 2-s − 4-s + 5-s − 7-s + 3·8-s − 10-s + 4·11-s − 2·13-s + 14-s − 16-s − 20-s − 4·22-s − 4·23-s + 25-s + 2·26-s + 28-s − 2·29-s − 8·31-s − 5·32-s − 35-s − 2·37-s + 3·40-s + 10·41-s − 4·44-s + 4·46-s + 49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.377·7-s + 1.06·8-s − 0.316·10-s + 1.20·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.223·20-s − 0.852·22-s − 0.834·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.371·29-s − 1.43·31-s − 0.883·32-s − 0.169·35-s − 0.328·37-s + 0.474·40-s + 1.56·41-s − 0.603·44-s + 0.589·46-s + 1/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.16238034350486, −13.70950353982830, −13.04774220024248, −12.64098691705144, −12.26513436505748, −11.50455247848949, −11.08832743613860, −10.47391680503316, −9.987077142530846, −9.478502985319631, −9.264506460965162, −8.742465174387273, −8.209161063175267, −7.447012989348416, −7.232674994678299, −6.511827541623182, −5.883239270303899, −5.485450426980195, −4.688753210062517, −4.168230343876841, −3.707543931049542, −2.955954106609540, −2.033528972715618, −1.609597769564622, −0.7879699129334668, 0,
0.7879699129334668, 1.609597769564622, 2.033528972715618, 2.955954106609540, 3.707543931049542, 4.168230343876841, 4.688753210062517, 5.485450426980195, 5.883239270303899, 6.511827541623182, 7.232674994678299, 7.447012989348416, 8.209161063175267, 8.742465174387273, 9.264506460965162, 9.478502985319631, 9.987077142530846, 10.47391680503316, 11.08832743613860, 11.50455247848949, 12.26513436505748, 12.64098691705144, 13.04774220024248, 13.70950353982830, 14.16238034350486
