| L(s) = 1 | − 2-s + 3·3-s − 4-s − 5-s − 3·6-s − 7-s + 3·8-s + 6·9-s + 10-s − 11-s − 3·12-s − 2·13-s + 14-s − 3·15-s − 16-s + 3·17-s − 6·18-s + 4·19-s + 20-s − 3·21-s + 22-s + 9·24-s + 25-s + 2·26-s + 9·27-s + 28-s + 10·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 1/2·4-s − 0.447·5-s − 1.22·6-s − 0.377·7-s + 1.06·8-s + 2·9-s + 0.316·10-s − 0.301·11-s − 0.866·12-s − 0.554·13-s + 0.267·14-s − 0.774·15-s − 1/4·16-s + 0.727·17-s − 1.41·18-s + 0.917·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s + 1.83·24-s + 1/5·25-s + 0.392·26-s + 1.73·27-s + 0.188·28-s + 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18515 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18515 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.373418184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.373418184\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−15.83751759716204, −15.22253892106027, −14.45610136948231, −14.20300308187880, −13.75063673276947, −13.12944991598451, −12.55813372352819, −12.13153185516502, −11.15645605126262, −10.32673097944919, −10.00977032590697, −9.445857156424017, −8.986014396559306, −8.471686393517889, −7.849440734397700, −7.521533808395293, −7.064458975596105, −5.966060466262354, −5.014610053759698, −4.377581598993532, −3.797397619023154, −2.992862742168567, −2.595668857876958, −1.481845998718128, −0.7047538416924796,
0.7047538416924796, 1.481845998718128, 2.595668857876958, 2.992862742168567, 3.797397619023154, 4.377581598993532, 5.014610053759698, 5.966060466262354, 7.064458975596105, 7.521533808395293, 7.849440734397700, 8.471686393517889, 8.986014396559306, 9.445857156424017, 10.00977032590697, 10.32673097944919, 11.15645605126262, 12.13153185516502, 12.55813372352819, 13.12944991598451, 13.75063673276947, 14.20300308187880, 14.45610136948231, 15.22253892106027, 15.83751759716204
