L(s) = 1 | + 2-s + 3-s + 4-s + 3.88·5-s + 6-s − 1.31·7-s + 8-s + 9-s + 3.88·10-s − 0.382·11-s + 12-s − 5.18·13-s − 1.31·14-s + 3.88·15-s + 16-s − 17-s + 18-s + 7.41·19-s + 3.88·20-s − 1.31·21-s − 0.382·22-s + 5.18·23-s + 24-s + 10.0·25-s − 5.18·26-s + 27-s − 1.31·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.73·5-s + 0.408·6-s − 0.496·7-s + 0.353·8-s + 0.333·9-s + 1.22·10-s − 0.115·11-s + 0.288·12-s − 1.43·13-s − 0.351·14-s + 1.00·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 1.70·19-s + 0.868·20-s − 0.286·21-s − 0.0816·22-s + 1.08·23-s + 0.204·24-s + 2.01·25-s − 1.01·26-s + 0.192·27-s − 0.248·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.461190299\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.461190299\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 - 3.88T + 5T^{2} \) |
| 7 | \( 1 + 1.31T + 7T^{2} \) |
| 11 | \( 1 + 0.382T + 11T^{2} \) |
| 13 | \( 1 + 5.18T + 13T^{2} \) |
| 19 | \( 1 - 7.41T + 19T^{2} \) |
| 23 | \( 1 - 5.18T + 23T^{2} \) |
| 29 | \( 1 - 2.99T + 29T^{2} \) |
| 31 | \( 1 + 5.17T + 31T^{2} \) |
| 37 | \( 1 + 6.75T + 37T^{2} \) |
| 41 | \( 1 - 8.40T + 41T^{2} \) |
| 43 | \( 1 - 7.41T + 43T^{2} \) |
| 47 | \( 1 - 6.28T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 8.74T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 9.21T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.919652746395077647511569818935, −7.07975565342563770166924892426, −6.76930864238178640323921585045, −5.69518045789920359267134288149, −5.32027775279014452127418859444, −4.62437839905934813611765690375, −3.42047921699697407571735788649, −2.70970057917732146657526204829, −2.21297837280853149736232207829, −1.13818569483158793239177409395,
1.13818569483158793239177409395, 2.21297837280853149736232207829, 2.70970057917732146657526204829, 3.42047921699697407571735788649, 4.62437839905934813611765690375, 5.32027775279014452127418859444, 5.69518045789920359267134288149, 6.76930864238178640323921585045, 7.07975565342563770166924892426, 7.919652746395077647511569818935