L(s) = 1 | + 2-s + 3-s + 4-s + 0.140·5-s + 6-s − 1.43·7-s + 8-s + 9-s + 0.140·10-s − 1.70·11-s + 12-s − 5.85·13-s − 1.43·14-s + 0.140·15-s + 16-s + 17-s + 18-s + 1.78·19-s + 0.140·20-s − 1.43·21-s − 1.70·22-s − 0.440·23-s + 24-s − 4.98·25-s − 5.85·26-s + 27-s − 1.43·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.0629·5-s + 0.408·6-s − 0.542·7-s + 0.353·8-s + 0.333·9-s + 0.0445·10-s − 0.514·11-s + 0.288·12-s − 1.62·13-s − 0.383·14-s + 0.0363·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.408·19-s + 0.0314·20-s − 0.313·21-s − 0.363·22-s − 0.0919·23-s + 0.204·24-s − 0.996·25-s − 1.14·26-s + 0.192·27-s − 0.271·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 - 0.140T + 5T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 + 5.85T + 13T^{2} \) |
| 19 | \( 1 - 1.78T + 19T^{2} \) |
| 23 | \( 1 + 0.440T + 23T^{2} \) |
| 29 | \( 1 - 5.52T + 29T^{2} \) |
| 31 | \( 1 + 9.59T + 31T^{2} \) |
| 37 | \( 1 - 6.81T + 37T^{2} \) |
| 41 | \( 1 + 3.27T + 41T^{2} \) |
| 43 | \( 1 + 3.73T + 43T^{2} \) |
| 47 | \( 1 + 1.32T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 61 | \( 1 + 2.22T + 61T^{2} \) |
| 67 | \( 1 + 8.22T + 67T^{2} \) |
| 71 | \( 1 + 0.429T + 71T^{2} \) |
| 73 | \( 1 + 1.75T + 73T^{2} \) |
| 79 | \( 1 + 0.426T + 79T^{2} \) |
| 83 | \( 1 - 7.32T + 83T^{2} \) |
| 89 | \( 1 - 4.66T + 89T^{2} \) |
| 97 | \( 1 + 19.0T + 97T^{2} \) |
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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
−7.66038774043089723523027445019, −7.05851400901191890575237223527, −6.29784642921380169289278339571, −5.42928886882668689513875945885, −4.84783733682756515691914729508, −4.00897046734707607657760374123, −3.13773312599722989313129477239, −2.58807961133991637383881104620, −1.67876378434921257480705756090, 0,
1.67876378434921257480705756090, 2.58807961133991637383881104620, 3.13773312599722989313129477239, 4.00897046734707607657760374123, 4.84783733682756515691914729508, 5.42928886882668689513875945885, 6.29784642921380169289278339571, 7.05851400901191890575237223527, 7.66038774043089723523027445019