| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s + 7-s − 8-s + 9-s + 2·10-s − 4·11-s + 12-s + 2·13-s − 14-s − 2·15-s + 16-s − 6·17-s − 18-s − 2·20-s + 21-s + 4·22-s − 23-s − 24-s − 25-s − 2·26-s + 27-s + 28-s + 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.447·20-s + 0.218·21-s + 0.852·22-s − 0.208·23-s − 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.17034741803910, −12.40773546961804, −12.06261623954654, −11.55206549819575, −11.10187983035216, −10.66647832976814, −10.44774750574568, −9.670529212732078, −9.416301021200528, −8.693296133370262, −8.333327230637152, −8.110636550702043, −7.709094212883558, −7.141799217607515, −6.666776307535916, −6.215231603106071, −5.549896766968571, −4.878579873320890, −4.441839865569226, −4.030510052660080, −3.258179555369229, −2.861708144860395, −2.339225618162108, −1.689855292237273, −1.131963019515052, 0, 0,
1.131963019515052, 1.689855292237273, 2.339225618162108, 2.861708144860395, 3.258179555369229, 4.030510052660080, 4.441839865569226, 4.878579873320890, 5.549896766968571, 6.215231603106071, 6.666776307535916, 7.141799217607515, 7.709094212883558, 8.110636550702043, 8.333327230637152, 8.693296133370262, 9.416301021200528, 9.670529212732078, 10.44774750574568, 10.66647832976814, 11.10187983035216, 11.55206549819575, 12.06261623954654, 12.40773546961804, 13.17034741803910
