| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2.38·7-s − 8-s + 9-s − 5.85·11-s + 12-s + 3.38·13-s − 2.38·14-s + 16-s + 7.70·17-s − 18-s + 4.09·19-s + 2.38·21-s + 5.85·22-s − 3.09·23-s − 24-s − 3.38·26-s + 27-s + 2.38·28-s + 5.70·29-s − 6.47·31-s − 32-s − 5.85·33-s − 7.70·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.900·7-s − 0.353·8-s + 0.333·9-s − 1.76·11-s + 0.288·12-s + 0.937·13-s − 0.636·14-s + 0.250·16-s + 1.86·17-s − 0.235·18-s + 0.938·19-s + 0.519·21-s + 1.24·22-s − 0.644·23-s − 0.204·24-s − 0.663·26-s + 0.192·27-s + 0.450·28-s + 1.05·29-s − 1.16·31-s − 0.176·32-s − 1.01·33-s − 1.32·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.499639419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.499639419\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.38T + 7T^{2} \) |
| 11 | \( 1 + 5.85T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 - 7.70T + 17T^{2} \) |
| 19 | \( 1 - 4.09T + 19T^{2} \) |
| 23 | \( 1 + 3.09T + 23T^{2} \) |
| 29 | \( 1 - 5.70T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 1.61T + 37T^{2} \) |
| 41 | \( 1 - 0.381T + 41T^{2} \) |
| 43 | \( 1 - 7.70T + 43T^{2} \) |
| 47 | \( 1 - 8.61T + 47T^{2} \) |
| 53 | \( 1 - 0.381T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 3.23T + 67T^{2} \) |
| 71 | \( 1 + 4.47T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−10.36159654423022920317707907717, −9.493608215871121679620853072776, −8.461571378487726807263584562430, −7.84462332378714446137790700393, −7.42870943787904088195404995620, −5.84832147928909660512638598894, −5.10886372119344622084759441837, −3.59445857792095036750700771086, −2.54159153840049157841681421308, −1.21254035295304341454546610332,
1.21254035295304341454546610332, 2.54159153840049157841681421308, 3.59445857792095036750700771086, 5.10886372119344622084759441837, 5.84832147928909660512638598894, 7.42870943787904088195404995620, 7.84462332378714446137790700393, 8.461571378487726807263584562430, 9.493608215871121679620853072776, 10.36159654423022920317707907717
