| L(s) = 1 | − 3-s − 5-s + 3·7-s + 9-s + 15-s − 17-s + 4·19-s − 3·21-s + 25-s − 27-s − 2·29-s − 3·35-s − 8·37-s − 2·41-s + 8·43-s − 45-s − 8·47-s + 2·49-s + 51-s − 4·53-s − 4·57-s + 5·59-s + 13·61-s + 3·63-s − 7·67-s − 9·71-s − 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.258·15-s − 0.242·17-s + 0.917·19-s − 0.654·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.507·35-s − 1.31·37-s − 0.312·41-s + 1.21·43-s − 0.149·45-s − 1.16·47-s + 2/7·49-s + 0.140·51-s − 0.549·53-s − 0.529·57-s + 0.650·59-s + 1.66·61-s + 0.377·63-s − 0.855·67-s − 1.06·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.036708349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.036708349\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−12.56718152942999, −11.79964342935474, −11.65265214609661, −11.40122801948019, −10.84028435323836, −10.29972743458320, −10.05097354338373, −9.321669896000170, −8.837226997664701, −8.459968908418436, −7.889656914331866, −7.430005221080080, −7.170571172797746, −6.520526427235691, −5.983201521677398, −5.370445571015945, −5.124414273447500, −4.505166898242083, −4.197079774999877, −3.403335174975870, −3.077186179783916, −2.085987351635488, −1.752350394626084, −1.039762884673886, −0.4331137327175294,
0.4331137327175294, 1.039762884673886, 1.752350394626084, 2.085987351635488, 3.077186179783916, 3.403335174975870, 4.197079774999877, 4.505166898242083, 5.124414273447500, 5.370445571015945, 5.983201521677398, 6.520526427235691, 7.170571172797746, 7.430005221080080, 7.889656914331866, 8.459968908418436, 8.837226997664701, 9.321669896000170, 10.05097354338373, 10.29972743458320, 10.84028435323836, 11.40122801948019, 11.65265214609661, 11.79964342935474, 12.56718152942999
