| L(s) = 1 | + 1.58·5-s + 1.11·7-s − 0.111·11-s − 0.888·13-s + 17-s − 19-s + 4.98·23-s − 2.47·25-s − 9.98·29-s − 1.51·31-s + 1.76·35-s − 3.69·37-s − 8.38·41-s − 4.98·43-s − 5.79·47-s − 5.76·49-s + 2.69·53-s − 0.176·55-s − 12.2·59-s − 8.28·61-s − 1.41·65-s + 12.6·67-s − 4.33·71-s − 3.60·73-s − 0.123·77-s + 2.74·79-s + 6.97·83-s + ⋯ |
| L(s) = 1 | + 0.710·5-s + 0.420·7-s − 0.0335·11-s − 0.246·13-s + 0.242·17-s − 0.229·19-s + 1.03·23-s − 0.495·25-s − 1.85·29-s − 0.271·31-s + 0.298·35-s − 0.608·37-s − 1.30·41-s − 0.760·43-s − 0.845·47-s − 0.823·49-s + 0.370·53-s − 0.0238·55-s − 1.59·59-s − 1.06·61-s − 0.175·65-s + 1.54·67-s − 0.514·71-s − 0.421·73-s − 0.0140·77-s + 0.308·79-s + 0.765·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.58T + 5T^{2} \) |
| 7 | \( 1 - 1.11T + 7T^{2} \) |
| 11 | \( 1 + 0.111T + 11T^{2} \) |
| 13 | \( 1 + 0.888T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 23 | \( 1 - 4.98T + 23T^{2} \) |
| 29 | \( 1 + 9.98T + 29T^{2} \) |
| 31 | \( 1 + 1.51T + 31T^{2} \) |
| 37 | \( 1 + 3.69T + 37T^{2} \) |
| 41 | \( 1 + 8.38T + 41T^{2} \) |
| 43 | \( 1 + 4.98T + 43T^{2} \) |
| 47 | \( 1 + 5.79T + 47T^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 8.28T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 4.33T + 71T^{2} \) |
| 73 | \( 1 + 3.60T + 73T^{2} \) |
| 79 | \( 1 - 2.74T + 79T^{2} \) |
| 83 | \( 1 - 6.97T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−7.52953915775685044470046877348, −6.72891264475015109519843448839, −6.10689205710971971468104957582, −5.22987695437916724385374473491, −4.93898315353814507476687996295, −3.80544199528596568579394978248, −3.11847222779783492775044742884, −2.02580115326791838275075771965, −1.50508081005203784002152917990, 0,
1.50508081005203784002152917990, 2.02580115326791838275075771965, 3.11847222779783492775044742884, 3.80544199528596568579394978248, 4.93898315353814507476687996295, 5.22987695437916724385374473491, 6.10689205710971971468104957582, 6.72891264475015109519843448839, 7.52953915775685044470046877348
