| L(s) = 1 | − 1.16·2-s − 3.26·3-s − 0.650·4-s + 3.42·5-s + 3.79·6-s + 3.07·8-s + 7.65·9-s − 3.98·10-s − 2.38·11-s + 2.12·12-s − 5.65·13-s − 11.1·15-s − 2.27·16-s − 5.86·17-s − 8.89·18-s − 0.160·19-s − 2.22·20-s + 2.77·22-s − 1.48·23-s − 10.0·24-s + 6.74·25-s + 6.56·26-s − 15.1·27-s + 8.05·29-s + 12.9·30-s + 6.15·31-s − 3.51·32-s + ⋯ |
| L(s) = 1 | − 0.821·2-s − 1.88·3-s − 0.325·4-s + 1.53·5-s + 1.54·6-s + 1.08·8-s + 2.55·9-s − 1.25·10-s − 0.720·11-s + 0.612·12-s − 1.56·13-s − 2.88·15-s − 0.569·16-s − 1.42·17-s − 2.09·18-s − 0.0369·19-s − 0.498·20-s + 0.591·22-s − 0.310·23-s − 2.05·24-s + 1.34·25-s + 1.28·26-s − 2.92·27-s + 1.49·29-s + 2.37·30-s + 1.10·31-s − 0.621·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4068155978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4068155978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 1.16T + 2T^{2} \) |
| 3 | \( 1 + 3.26T + 3T^{2} \) |
| 5 | \( 1 - 3.42T + 5T^{2} \) |
| 11 | \( 1 + 2.38T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 + 0.160T + 19T^{2} \) |
| 23 | \( 1 + 1.48T + 23T^{2} \) |
| 29 | \( 1 - 8.05T + 29T^{2} \) |
| 31 | \( 1 - 6.15T + 31T^{2} \) |
| 37 | \( 1 + 8.65T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 4.26T + 43T^{2} \) |
| 47 | \( 1 + 4.61T + 47T^{2} \) |
| 53 | \( 1 - 1.20T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 3.10T + 61T^{2} \) |
| 67 | \( 1 - 4.72T + 67T^{2} \) |
| 71 | \( 1 - 0.550T + 71T^{2} \) |
| 73 | \( 1 + 8.39T + 73T^{2} \) |
| 83 | \( 1 + 1.63T + 83T^{2} \) |
| 89 | \( 1 - 0.156T + 89T^{2} \) |
| 97 | \( 1 - 0.950T + 97T^{2} \) |
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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
−8.654518111390058148855386673140, −7.60796629973447226703571708684, −6.85161417968590960670786122175, −6.29768354159372046372444751927, −5.49398980750596950466387922000, −4.76652814693066004844563263305, −4.57112689153416955105696370614, −2.48217555250833190101642271014, −1.62544681238781910688046879351, −0.46820581961916486149658617143,
0.46820581961916486149658617143, 1.62544681238781910688046879351, 2.48217555250833190101642271014, 4.57112689153416955105696370614, 4.76652814693066004844563263305, 5.49398980750596950466387922000, 6.29768354159372046372444751927, 6.85161417968590960670786122175, 7.60796629973447226703571708684, 8.654518111390058148855386673140
