L(s) = 1 | + 2-s − 3-s + 4-s + 1.06·5-s − 6-s − 4.42·7-s + 8-s + 9-s + 1.06·10-s + 1.64·11-s − 12-s + 5.47·13-s − 4.42·14-s − 1.06·15-s + 16-s − 3.92·17-s + 18-s − 6.60·19-s + 1.06·20-s + 4.42·21-s + 1.64·22-s − 24-s − 3.86·25-s + 5.47·26-s − 27-s − 4.42·28-s − 6.39·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.477·5-s − 0.408·6-s − 1.67·7-s + 0.353·8-s + 0.333·9-s + 0.337·10-s + 0.497·11-s − 0.288·12-s + 1.51·13-s − 1.18·14-s − 0.275·15-s + 0.250·16-s − 0.952·17-s + 0.235·18-s − 1.51·19-s + 0.238·20-s + 0.965·21-s + 0.351·22-s − 0.204·24-s − 0.772·25-s + 1.07·26-s − 0.192·27-s − 0.836·28-s − 1.18·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - 1.06T + 5T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 + 3.92T + 17T^{2} \) |
| 19 | \( 1 + 6.60T + 19T^{2} \) |
| 29 | \( 1 + 6.39T + 29T^{2} \) |
| 31 | \( 1 + 3.01T + 31T^{2} \) |
| 37 | \( 1 - 2.21T + 37T^{2} \) |
| 41 | \( 1 - 4.09T + 41T^{2} \) |
| 43 | \( 1 - 2.91T + 43T^{2} \) |
| 47 | \( 1 + 9.62T + 47T^{2} \) |
| 53 | \( 1 - 2.23T + 53T^{2} \) |
| 59 | \( 1 - 9.75T + 59T^{2} \) |
| 61 | \( 1 - 1.55T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 5.26T + 71T^{2} \) |
| 73 | \( 1 - 7.18T + 73T^{2} \) |
| 79 | \( 1 + 2.69T + 79T^{2} \) |
| 83 | \( 1 + 5.53T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.377725481523399297064329541475, −7.11339622014463365530265713708, −6.45553879783222072377231845182, −6.13047860758635256558406462519, −5.51029943219125267284056393066, −4.07793060246458039053185437910, −3.87614770639321064270526402425, −2.68138003164977174830814356784, −1.61373198798947695845898684083, 0,
1.61373198798947695845898684083, 2.68138003164977174830814356784, 3.87614770639321064270526402425, 4.07793060246458039053185437910, 5.51029943219125267284056393066, 6.13047860758635256558406462519, 6.45553879783222072377231845182, 7.11339622014463365530265713708, 8.377725481523399297064329541475