| L(s) = 1 | + 2.38·2-s − 3-s + 3.69·4-s + 1.78·5-s − 2.38·6-s + 3.39·7-s + 4.04·8-s + 9-s + 4.24·10-s − 11-s − 3.69·12-s + 1.16·13-s + 8.10·14-s − 1.78·15-s + 2.26·16-s + 2.82·17-s + 2.38·18-s + 0.909·19-s + 6.58·20-s − 3.39·21-s − 2.38·22-s + 5.05·23-s − 4.04·24-s − 1.82·25-s + 2.79·26-s − 27-s + 12.5·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 0.577·3-s + 1.84·4-s + 0.796·5-s − 0.974·6-s + 1.28·7-s + 1.43·8-s + 0.333·9-s + 1.34·10-s − 0.301·11-s − 1.06·12-s + 0.324·13-s + 2.16·14-s − 0.459·15-s + 0.566·16-s + 0.684·17-s + 0.562·18-s + 0.208·19-s + 1.47·20-s − 0.741·21-s − 0.508·22-s + 1.05·23-s − 0.825·24-s − 0.365·25-s + 0.547·26-s − 0.192·27-s + 2.37·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.877391939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.877391939\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 2 | \( 1 - 2.38T + 2T^{2} \) |
| 5 | \( 1 - 1.78T + 5T^{2} \) |
| 7 | \( 1 - 3.39T + 7T^{2} \) |
| 13 | \( 1 - 1.16T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 0.909T + 19T^{2} \) |
| 23 | \( 1 - 5.05T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 + 4.28T + 31T^{2} \) |
| 37 | \( 1 + 6.75T + 37T^{2} \) |
| 41 | \( 1 - 2.84T + 41T^{2} \) |
| 43 | \( 1 - 2.41T + 43T^{2} \) |
| 47 | \( 1 - 1.46T + 47T^{2} \) |
| 53 | \( 1 - 8.03T + 53T^{2} \) |
| 59 | \( 1 + 3.17T + 59T^{2} \) |
| 61 | \( 1 + 6.04T + 61T^{2} \) |
| 67 | \( 1 + 4.12T + 67T^{2} \) |
| 71 | \( 1 - 5.55T + 71T^{2} \) |
| 73 | \( 1 - 8.74T + 73T^{2} \) |
| 79 | \( 1 + 4.66T + 79T^{2} \) |
| 83 | \( 1 + 8.63T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
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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
−8.560859179762535882515902256352, −7.55039407660176248501235279475, −6.94055809887221488409220862479, −5.95207068424130252772725985289, −5.49096560847855684660645589496, −4.98702031661156027129866670880, −4.22069726272841269641872912304, −3.25413724810085583801029380851, −2.19900181684320349936253616577, −1.35255240156070171321587342942,
1.35255240156070171321587342942, 2.19900181684320349936253616577, 3.25413724810085583801029380851, 4.22069726272841269641872912304, 4.98702031661156027129866670880, 5.49096560847855684660645589496, 5.95207068424130252772725985289, 6.94055809887221488409220862479, 7.55039407660176248501235279475, 8.560859179762535882515902256352
