| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 2·13-s + 16-s + 2·17-s + 19-s + 20-s − 4·23-s + 25-s − 2·26-s + 6·29-s + 4·31-s + 32-s + 2·34-s − 6·37-s + 38-s + 40-s + 10·41-s + 4·43-s − 4·46-s + 12·47-s − 7·49-s + 50-s − 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.229·19-s + 0.223·20-s − 0.834·23-s + 1/5·25-s − 0.392·26-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.342·34-s − 0.986·37-s + 0.162·38-s + 0.158·40-s + 1.56·41-s + 0.609·43-s − 0.589·46-s + 1.75·47-s − 49-s + 0.141·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.327946752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.327946752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.96117776849731, −12.56437653738391, −12.18489277657981, −11.79328599178268, −11.22089751750414, −10.71488441145248, −10.20348329372198, −9.847922354417577, −9.409373064055916, −8.709394483280260, −8.249316974425508, −7.723931956622556, −7.162221077690059, −6.803733664992239, −6.044331394232127, −5.854601527172320, −5.233829843661782, −4.712989350339354, −4.230682777351579, −3.674411300426389, −2.977000614222208, −2.530076267366002, −2.010297850369985, −1.222589074168782, −0.5900776591449272,
0.5900776591449272, 1.222589074168782, 2.010297850369985, 2.530076267366002, 2.977000614222208, 3.674411300426389, 4.230682777351579, 4.712989350339354, 5.233829843661782, 5.854601527172320, 6.044331394232127, 6.803733664992239, 7.162221077690059, 7.723931956622556, 8.249316974425508, 8.709394483280260, 9.409373064055916, 9.847922354417577, 10.20348329372198, 10.71488441145248, 11.22089751750414, 11.79328599178268, 12.18489277657981, 12.56437653738391, 12.96117776849731
