L(s) = 1 | + 5·5-s − 20·7-s − 24·11-s + 74·13-s − 54·17-s + 124·19-s − 120·23-s + 25·25-s + 78·29-s − 200·31-s − 100·35-s − 70·37-s − 330·41-s − 92·43-s − 24·47-s + 57·49-s − 450·53-s − 120·55-s + 24·59-s − 322·61-s + 370·65-s + 196·67-s − 288·71-s − 430·73-s + 480·77-s + 520·79-s + 156·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.07·7-s − 0.657·11-s + 1.57·13-s − 0.770·17-s + 1.49·19-s − 1.08·23-s + 1/5·25-s + 0.499·29-s − 1.15·31-s − 0.482·35-s − 0.311·37-s − 1.25·41-s − 0.326·43-s − 0.0744·47-s + 0.166·49-s − 1.16·53-s − 0.294·55-s + 0.0529·59-s − 0.675·61-s + 0.706·65-s + 0.357·67-s − 0.481·71-s − 0.689·73-s + 0.710·77-s + 0.740·79-s + 0.206·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
good | 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 74 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 78 T + p^{3} T^{2} \) |
| 31 | \( 1 + 200 T + p^{3} T^{2} \) |
| 37 | \( 1 + 70 T + p^{3} T^{2} \) |
| 41 | \( 1 + 330 T + p^{3} T^{2} \) |
| 43 | \( 1 + 92 T + p^{3} T^{2} \) |
| 47 | \( 1 + 24 T + p^{3} T^{2} \) |
| 53 | \( 1 + 450 T + p^{3} T^{2} \) |
| 59 | \( 1 - 24 T + p^{3} T^{2} \) |
| 61 | \( 1 + 322 T + p^{3} T^{2} \) |
| 67 | \( 1 - 196 T + p^{3} T^{2} \) |
| 71 | \( 1 + 288 T + p^{3} T^{2} \) |
| 73 | \( 1 + 430 T + p^{3} T^{2} \) |
| 79 | \( 1 - 520 T + p^{3} T^{2} \) |
| 83 | \( 1 - 156 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1026 T + p^{3} T^{2} \) |
| 97 | \( 1 + 286 T + p^{3} T^{2} \) |
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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
−9.613143644582392886717763762776, −8.839451752879676223134202541239, −7.897407590603166405793701936968, −6.76700048975469624763125675027, −6.07678816660752157406032692866, −5.20578103779215622029772210758, −3.78942925514281055842750797099, −2.96644104411721720941066105704, −1.55152849596739807232836114335, 0,
1.55152849596739807232836114335, 2.96644104411721720941066105704, 3.78942925514281055842750797099, 5.20578103779215622029772210758, 6.07678816660752157406032692866, 6.76700048975469624763125675027, 7.897407590603166405793701936968, 8.839451752879676223134202541239, 9.613143644582392886717763762776