| L(s) = 1 | − 3-s − 1.03·5-s + 0.840·7-s + 9-s − 2.75·11-s − 2.29·13-s + 1.03·15-s − 4.89·17-s + 5.97·19-s − 0.840·21-s − 23-s − 3.92·25-s − 27-s − 29-s + 2.67·31-s + 2.75·33-s − 0.871·35-s − 3.96·37-s + 2.29·39-s + 0.116·41-s − 11.4·43-s − 1.03·45-s + 10.0·47-s − 6.29·49-s + 4.89·51-s + 3.27·53-s + 2.85·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.463·5-s + 0.317·7-s + 0.333·9-s − 0.829·11-s − 0.635·13-s + 0.267·15-s − 1.18·17-s + 1.37·19-s − 0.183·21-s − 0.208·23-s − 0.784·25-s − 0.192·27-s − 0.185·29-s + 0.480·31-s + 0.479·33-s − 0.147·35-s − 0.652·37-s + 0.366·39-s + 0.0181·41-s − 1.74·43-s − 0.154·45-s + 1.46·47-s − 0.899·49-s + 0.684·51-s + 0.449·53-s + 0.384·55-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
| \(L(1)\) |
\(\approx\) |
\(0.8712429599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8712429599\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.03T + 5T^{2} \) |
| 7 | \( 1 - 0.840T + 7T^{2} \) |
| 11 | \( 1 + 2.75T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 - 5.97T + 19T^{2} \) |
| 31 | \( 1 - 2.67T + 31T^{2} \) |
| 37 | \( 1 + 3.96T + 37T^{2} \) |
| 41 | \( 1 - 0.116T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 3.27T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 - 5.07T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 9.22T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 7.23T + 79T^{2} \) |
| 83 | \( 1 + 9.99T + 83T^{2} \) |
| 89 | \( 1 - 8.41T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.67390328724972001798492795243, −7.24885374010703771148309247851, −6.50773931819056839359039931904, −5.59500541967961048881943406799, −5.08138080314573433733343864141, −4.43024052826959269644605803558, −3.58978075105263832305496103368, −2.63592776239341607583516479700, −1.75325112959105992341309139325, −0.46513015885722999411061165664,
0.46513015885722999411061165664, 1.75325112959105992341309139325, 2.63592776239341607583516479700, 3.58978075105263832305496103368, 4.43024052826959269644605803558, 5.08138080314573433733343864141, 5.59500541967961048881943406799, 6.50773931819056839359039931904, 7.24885374010703771148309247851, 7.67390328724972001798492795243
