| L(s) = 1 | − 4·3-s − 5·5-s − 11·9-s − 60·11-s − 86·13-s + 20·15-s − 18·17-s − 44·19-s + 48·23-s + 25·25-s + 152·27-s − 186·29-s − 176·31-s + 240·33-s + 254·37-s + 344·39-s − 186·41-s − 100·43-s + 55·45-s − 168·47-s + 72·51-s − 498·53-s + 300·55-s + 176·57-s + 252·59-s + 58·61-s + 430·65-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.447·5-s − 0.407·9-s − 1.64·11-s − 1.83·13-s + 0.344·15-s − 0.256·17-s − 0.531·19-s + 0.435·23-s + 1/5·25-s + 1.08·27-s − 1.19·29-s − 1.01·31-s + 1.26·33-s + 1.12·37-s + 1.41·39-s − 0.708·41-s − 0.354·43-s + 0.182·45-s − 0.521·47-s + 0.197·51-s − 1.29·53-s + 0.735·55-s + 0.408·57-s + 0.556·59-s + 0.121·61-s + 0.820·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1548948953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1548948953\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 86 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 186 T + p^{3} T^{2} \) |
| 31 | \( 1 + 176 T + p^{3} T^{2} \) |
| 37 | \( 1 - 254 T + p^{3} T^{2} \) |
| 41 | \( 1 + 186 T + p^{3} T^{2} \) |
| 43 | \( 1 + 100 T + p^{3} T^{2} \) |
| 47 | \( 1 + 168 T + p^{3} T^{2} \) |
| 53 | \( 1 + 498 T + p^{3} T^{2} \) |
| 59 | \( 1 - 252 T + p^{3} T^{2} \) |
| 61 | \( 1 - 58 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1036 T + p^{3} T^{2} \) |
| 71 | \( 1 - 168 T + p^{3} T^{2} \) |
| 73 | \( 1 + 506 T + p^{3} T^{2} \) |
| 79 | \( 1 - 272 T + p^{3} T^{2} \) |
| 83 | \( 1 + 948 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1014 T + p^{3} T^{2} \) |
| 97 | \( 1 - 766 T + p^{3} T^{2} \) |
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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
−9.782773015162915816757264780628, −8.750891333043183843279610011035, −7.74107284875093723725016957420, −7.24449246823445332753133006130, −6.06154538291606579055345712367, −5.16127942289439250077705481005, −4.67357550709088190785833343206, −3.13446257455874848327095497938, −2.19987909034945642170062416093, −0.20206133059242065576440346648,
0.20206133059242065576440346648, 2.19987909034945642170062416093, 3.13446257455874848327095497938, 4.67357550709088190785833343206, 5.16127942289439250077705481005, 6.06154538291606579055345712367, 7.24449246823445332753133006130, 7.74107284875093723725016957420, 8.750891333043183843279610011035, 9.782773015162915816757264780628
