| L(s) = 1 | + 2-s + 3-s + 4-s − 3.93·5-s + 6-s + 8-s + 9-s − 3.93·10-s + 1.27·11-s + 12-s − 5.94·13-s − 3.93·15-s + 16-s + 3.05·17-s + 18-s + 19-s − 3.93·20-s + 1.27·22-s − 0.148·23-s + 24-s + 10.5·25-s − 5.94·26-s + 27-s − 6.60·29-s − 3.93·30-s − 8.01·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.76·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.24·10-s + 0.383·11-s + 0.288·12-s − 1.64·13-s − 1.01·15-s + 0.250·16-s + 0.740·17-s + 0.235·18-s + 0.229·19-s − 0.880·20-s + 0.271·22-s − 0.0310·23-s + 0.204·24-s + 2.10·25-s − 1.16·26-s + 0.192·27-s − 1.22·29-s − 0.719·30-s − 1.44·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.449256437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.449256437\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 3.93T + 5T^{2} \) |
| 11 | \( 1 - 1.27T + 11T^{2} \) |
| 13 | \( 1 + 5.94T + 13T^{2} \) |
| 17 | \( 1 - 3.05T + 17T^{2} \) |
| 23 | \( 1 + 0.148T + 23T^{2} \) |
| 29 | \( 1 + 6.60T + 29T^{2} \) |
| 31 | \( 1 + 8.01T + 31T^{2} \) |
| 37 | \( 1 - 6.14T + 37T^{2} \) |
| 41 | \( 1 - 6.28T + 41T^{2} \) |
| 43 | \( 1 - 7.91T + 43T^{2} \) |
| 47 | \( 1 - 8.22T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 1.84T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 2.61T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 - 7.44T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 5.40T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−7.74716642423182887666805648262, −7.42836202397045905978954664895, −7.15087378640636091460110725580, −5.84712139485989907314141946622, −5.08378340055713581764768858049, −4.17094732772890292798349515056, −3.87986242604996688994771922204, −3.00729299538263517718740552959, −2.21338014595964281463971034275, −0.71257443160209577556222290161,
0.71257443160209577556222290161, 2.21338014595964281463971034275, 3.00729299538263517718740552959, 3.87986242604996688994771922204, 4.17094732772890292798349515056, 5.08378340055713581764768858049, 5.84712139485989907314141946622, 7.15087378640636091460110725580, 7.42836202397045905978954664895, 7.74716642423182887666805648262
