| L(s) = 1 | − 5·2-s + 17·4-s + 5·5-s − 45·8-s − 25·10-s − 12·11-s − 30·13-s + 89·16-s − 134·17-s + 92·19-s + 85·20-s + 60·22-s − 112·23-s + 25·25-s + 150·26-s + 58·29-s + 224·31-s − 85·32-s + 670·34-s − 146·37-s − 460·38-s − 225·40-s + 18·41-s + 340·43-s − 204·44-s + 560·46-s + 208·47-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 17/8·4-s + 0.447·5-s − 1.98·8-s − 0.790·10-s − 0.328·11-s − 0.640·13-s + 1.39·16-s − 1.91·17-s + 1.11·19-s + 0.950·20-s + 0.581·22-s − 1.01·23-s + 1/5·25-s + 1.13·26-s + 0.371·29-s + 1.29·31-s − 0.469·32-s + 3.37·34-s − 0.648·37-s − 1.96·38-s − 0.889·40-s + 0.0685·41-s + 1.20·43-s − 0.698·44-s + 1.79·46-s + 0.645·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 30 T + p^{3} T^{2} \) |
| 17 | \( 1 + 134 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 112 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 224 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 18 T + p^{3} T^{2} \) |
| 43 | \( 1 - 340 T + p^{3} T^{2} \) |
| 47 | \( 1 - 208 T + p^{3} T^{2} \) |
| 53 | \( 1 - 754 T + p^{3} T^{2} \) |
| 59 | \( 1 - 380 T + p^{3} T^{2} \) |
| 61 | \( 1 + 718 T + p^{3} T^{2} \) |
| 67 | \( 1 - 412 T + p^{3} T^{2} \) |
| 71 | \( 1 - 960 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1066 T + p^{3} T^{2} \) |
| 79 | \( 1 - 896 T + p^{3} T^{2} \) |
| 83 | \( 1 - 436 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 - 702 T + p^{3} T^{2} \) |
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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
−8.454518633409177646001690614370, −7.71382454533533323090619164101, −6.97601987220631453432172758318, −6.35940570680834285867658930978, −5.34970623623319497523255948157, −4.22629437241321382356219299771, −2.66196910331395592679785324185, −2.18193678356828036527579923922, −1.00192242183602422735834273132, 0,
1.00192242183602422735834273132, 2.18193678356828036527579923922, 2.66196910331395592679785324185, 4.22629437241321382356219299771, 5.34970623623319497523255948157, 6.35940570680834285867658930978, 6.97601987220631453432172758318, 7.71382454533533323090619164101, 8.454518633409177646001690614370
