| L(s) = 1 | − 2-s + 3-s + 4-s + 3.03·5-s − 6-s − 4.00·7-s − 8-s + 9-s − 3.03·10-s + 4.73·11-s + 12-s − 13-s + 4.00·14-s + 3.03·15-s + 16-s − 0.341·17-s − 18-s + 3.16·19-s + 3.03·20-s − 4.00·21-s − 4.73·22-s − 1.56·23-s − 24-s + 4.21·25-s + 26-s + 27-s − 4.00·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.35·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 0.333·9-s − 0.960·10-s + 1.42·11-s + 0.288·12-s − 0.277·13-s + 1.07·14-s + 0.783·15-s + 0.250·16-s − 0.0828·17-s − 0.235·18-s + 0.725·19-s + 0.678·20-s − 0.874·21-s − 1.00·22-s − 0.325·23-s − 0.204·24-s + 0.843·25-s + 0.196·26-s + 0.192·27-s − 0.756·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.295346320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.295346320\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 3.03T + 5T^{2} \) |
| 7 | \( 1 + 4.00T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 17 | \( 1 + 0.341T + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 + 1.56T + 23T^{2} \) |
| 29 | \( 1 - 3.07T + 29T^{2} \) |
| 31 | \( 1 - 5.42T + 31T^{2} \) |
| 37 | \( 1 + 2.78T + 37T^{2} \) |
| 41 | \( 1 + 4.18T + 41T^{2} \) |
| 43 | \( 1 - 7.70T + 43T^{2} \) |
| 47 | \( 1 - 1.39T + 47T^{2} \) |
| 53 | \( 1 + 3.73T + 53T^{2} \) |
| 59 | \( 1 + 0.728T + 59T^{2} \) |
| 61 | \( 1 - 7.75T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 0.572T + 71T^{2} \) |
| 73 | \( 1 - 9.69T + 73T^{2} \) |
| 79 | \( 1 - 4.97T + 79T^{2} \) |
| 83 | \( 1 - 9.54T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 0.676T + 97T^{2} \) |
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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
−7.87632901902976224431394073794, −7.02579249996250171357310415607, −6.42583204635778102567512470608, −6.17744632822447468977081398117, −5.21071770864631149004388828411, −4.06095970867176394768014049792, −3.25526659642271737557663145441, −2.59563219410655961230177441900, −1.75174283959727475803534278704, −0.830571139059607480452095805373,
0.830571139059607480452095805373, 1.75174283959727475803534278704, 2.59563219410655961230177441900, 3.25526659642271737557663145441, 4.06095970867176394768014049792, 5.21071770864631149004388828411, 6.17744632822447468977081398117, 6.42583204635778102567512470608, 7.02579249996250171357310415607, 7.87632901902976224431394073794
