| L(s) = 1 | − 0.605·3-s − 12.4·5-s − 0.605·7-s − 26.6·9-s − 57.0·11-s − 18.4·13-s + 7.52·15-s + 17·17-s − 68.4·19-s + 0.366·21-s − 53.4·23-s + 29.3·25-s + 32.4·27-s − 141.·29-s − 125.·31-s + 34.5·33-s + 7.52·35-s − 83.4·37-s + 11.1·39-s + 519.·41-s − 374.·43-s + 330.·45-s + 561.·47-s − 342.·49-s − 10.2·51-s − 562.·53-s + 709.·55-s + ⋯ |
| L(s) = 1 | − 0.116·3-s − 1.11·5-s − 0.0326·7-s − 0.986·9-s − 1.56·11-s − 0.394·13-s + 0.129·15-s + 0.242·17-s − 0.826·19-s + 0.00381·21-s − 0.484·23-s + 0.234·25-s + 0.231·27-s − 0.908·29-s − 0.729·31-s + 0.182·33-s + 0.0363·35-s − 0.370·37-s + 0.0459·39-s + 1.97·41-s − 1.32·43-s + 1.09·45-s + 1.74·47-s − 0.998·49-s − 0.0282·51-s − 1.45·53-s + 1.73·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3106325314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3106325314\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 + 0.605T + 27T^{2} \) |
| 5 | \( 1 + 12.4T + 125T^{2} \) |
| 7 | \( 1 + 0.605T + 343T^{2} \) |
| 11 | \( 1 + 57.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 18.4T + 2.19e3T^{2} \) |
| 19 | \( 1 + 68.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 53.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 141.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 125.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 83.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 519.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 374.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 561.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 562.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 246.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 291.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 729.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 911.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 644.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 533.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 51.9T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 178.T + 9.12e5T^{2} \) |
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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
−9.457347829340439189735513390868, −8.425964531736240783778075030482, −7.898557425107373525999776339309, −7.22466927321426147536114887113, −5.95588258068010235239491502151, −5.23161921104186510507664117838, −4.20703127715599649448053499044, −3.19674883982263547998859420490, −2.23433172029081943665123173484, −0.27220135418071016281284152171,
0.27220135418071016281284152171, 2.23433172029081943665123173484, 3.19674883982263547998859420490, 4.20703127715599649448053499044, 5.23161921104186510507664117838, 5.95588258068010235239491502151, 7.22466927321426147536114887113, 7.898557425107373525999776339309, 8.425964531736240783778075030482, 9.457347829340439189735513390868
