L(s) = 1 | − 3-s − 5-s − 2.91·7-s + 9-s − 3.43·11-s + 4.91·13-s + 15-s − 4.35·17-s + 19-s + 2.91·21-s − 6.35·23-s + 25-s − 27-s − 7.27·29-s + 2.35·31-s + 3.43·33-s + 2.91·35-s + 4.91·37-s − 4.91·39-s + 1.43·41-s − 6.91·43-s − 45-s + 6.35·47-s + 1.51·49-s + 4.35·51-s − 4.35·53-s + 3.43·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.10·7-s + 0.333·9-s − 1.03·11-s + 1.36·13-s + 0.258·15-s − 1.05·17-s + 0.229·19-s + 0.636·21-s − 1.32·23-s + 0.200·25-s − 0.192·27-s − 1.35·29-s + 0.422·31-s + 0.598·33-s + 0.493·35-s + 0.808·37-s − 0.787·39-s + 0.224·41-s − 1.05·43-s − 0.149·45-s + 0.926·47-s + 0.216·49-s + 0.609·51-s − 0.598·53-s + 0.463·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6422514274\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6422514274\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2.91T + 7T^{2} \) |
| 11 | \( 1 + 3.43T + 11T^{2} \) |
| 13 | \( 1 - 4.91T + 13T^{2} \) |
| 17 | \( 1 + 4.35T + 17T^{2} \) |
| 23 | \( 1 + 6.35T + 23T^{2} \) |
| 29 | \( 1 + 7.27T + 29T^{2} \) |
| 31 | \( 1 - 2.35T + 31T^{2} \) |
| 37 | \( 1 - 4.91T + 37T^{2} \) |
| 41 | \( 1 - 1.43T + 41T^{2} \) |
| 43 | \( 1 + 6.91T + 43T^{2} \) |
| 47 | \( 1 - 6.35T + 47T^{2} \) |
| 53 | \( 1 + 4.35T + 53T^{2} \) |
| 59 | \( 1 + 9.83T + 59T^{2} \) |
| 61 | \( 1 + 6.19T + 61T^{2} \) |
| 67 | \( 1 - 8.70T + 67T^{2} \) |
| 71 | \( 1 + 6.87T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 8.70T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 4.40T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.194349244145207073905482537800, −7.65573518970770487815204685208, −6.69937079791830339775835528432, −6.16938542931036424871859811240, −5.56715452265414334711730264005, −4.53121116695540434977518117393, −3.80262046352314532635788470346, −3.03650784455289186903572829244, −1.90334398580523753014845680352, −0.44156962705337997371582218644,
0.44156962705337997371582218644, 1.90334398580523753014845680352, 3.03650784455289186903572829244, 3.80262046352314532635788470346, 4.53121116695540434977518117393, 5.56715452265414334711730264005, 6.16938542931036424871859811240, 6.69937079791830339775835528432, 7.65573518970770487815204685208, 8.194349244145207073905482537800