| L(s) = 1 | + 3-s − 3.07·5-s − 0.993·7-s + 9-s + 0.404·11-s + 5.75·13-s − 3.07·15-s − 1.51·17-s + 4.89·19-s − 0.993·21-s − 23-s + 4.42·25-s + 27-s + 29-s − 0.870·31-s + 0.404·33-s + 3.05·35-s − 1.02·37-s + 5.75·39-s − 2.16·41-s + 7.95·43-s − 3.07·45-s + 0.0449·47-s − 6.01·49-s − 1.51·51-s + 3.90·53-s − 1.24·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.37·5-s − 0.375·7-s + 0.333·9-s + 0.121·11-s + 1.59·13-s − 0.792·15-s − 0.366·17-s + 1.12·19-s − 0.216·21-s − 0.208·23-s + 0.885·25-s + 0.192·27-s + 0.185·29-s − 0.156·31-s + 0.0703·33-s + 0.515·35-s − 0.168·37-s + 0.922·39-s − 0.338·41-s + 1.21·43-s − 0.457·45-s + 0.00655·47-s − 0.858·49-s − 0.211·51-s + 0.535·53-s − 0.167·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.865681620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.865681620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 3.07T + 5T^{2} \) |
| 7 | \( 1 + 0.993T + 7T^{2} \) |
| 11 | \( 1 - 0.404T + 11T^{2} \) |
| 13 | \( 1 - 5.75T + 13T^{2} \) |
| 17 | \( 1 + 1.51T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 31 | \( 1 + 0.870T + 31T^{2} \) |
| 37 | \( 1 + 1.02T + 37T^{2} \) |
| 41 | \( 1 + 2.16T + 41T^{2} \) |
| 43 | \( 1 - 7.95T + 43T^{2} \) |
| 47 | \( 1 - 0.0449T + 47T^{2} \) |
| 53 | \( 1 - 3.90T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 9.21T + 67T^{2} \) |
| 71 | \( 1 + 1.42T + 71T^{2} \) |
| 73 | \( 1 - 4.05T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + 3.79T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−7.78861344963283756003730473788, −7.39738800023306120645087740148, −6.50465502509936154518179014483, −5.90612316297006572550854277806, −4.80082198678318773837548748408, −4.10877125313425553507071350989, −3.45079737877949358601604641262, −3.04368296738885930782511919763, −1.70309913158125052026615094244, −0.67435049852345279114635042607,
0.67435049852345279114635042607, 1.70309913158125052026615094244, 3.04368296738885930782511919763, 3.45079737877949358601604641262, 4.10877125313425553507071350989, 4.80082198678318773837548748408, 5.90612316297006572550854277806, 6.50465502509936154518179014483, 7.39738800023306120645087740148, 7.78861344963283756003730473788
