| L(s) = 1 | − 1.41·2-s + 1.41·3-s + 1.00·4-s − 5-s − 2.00·6-s + 1.00·9-s + 1.41·10-s + 1.41·12-s − 1.41·13-s − 1.41·15-s − 0.999·16-s − 1.41·18-s − 19-s − 1.00·20-s + 25-s + 2.00·26-s + 2.00·30-s + 1.41·32-s + 1.00·36-s + 1.41·37-s + 1.41·38-s − 2.00·39-s − 1.00·45-s − 1.41·48-s + 49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.41·3-s + 1.00·4-s − 5-s − 2.00·6-s + 1.00·9-s + 1.41·10-s + 1.41·12-s − 1.41·13-s − 1.41·15-s − 0.999·16-s − 1.41·18-s − 19-s − 1.00·20-s + 25-s + 2.00·26-s + 2.00·30-s + 1.41·32-s + 1.00·36-s + 1.41·37-s + 1.41·38-s − 2.00·39-s − 1.00·45-s − 1.41·48-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3812190448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3812190448\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.41T + T^{2} \) |
| 3 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.41T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.41T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 1.41T + T^{2} \) |
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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
−14.61957954625341714164664019560, −13.27042701489436960383904648520, −11.97440438418630739376367126084, −10.64837931859429375334750285042, −9.566156585394324383794480118050, −8.691956235481506596317612803994, −7.88660575820453984407934978439, −7.14472277853861817875692871455, −4.29396961792893118367281974002, −2.51518215952811900062369427304,
2.51518215952811900062369427304, 4.29396961792893118367281974002, 7.14472277853861817875692871455, 7.88660575820453984407934978439, 8.691956235481506596317612803994, 9.566156585394324383794480118050, 10.64837931859429375334750285042, 11.97440438418630739376367126084, 13.27042701489436960383904648520, 14.61957954625341714164664019560
