| L(s) = 1 | + 2.57·2-s − 3.30·3-s + 4.61·4-s + 4.29·5-s − 8.50·6-s − 0.180·7-s + 6.73·8-s + 7.92·9-s + 11.0·10-s − 2.21·11-s − 15.2·12-s − 0.463·14-s − 14.1·15-s + 8.08·16-s + 3.03·17-s + 20.3·18-s − 1.10·19-s + 19.8·20-s + 0.595·21-s − 5.69·22-s + 2.45·23-s − 22.2·24-s + 13.4·25-s − 16.2·27-s − 0.831·28-s − 5.06·29-s − 36.4·30-s + ⋯ |
| L(s) = 1 | + 1.81·2-s − 1.90·3-s + 2.30·4-s + 1.92·5-s − 3.47·6-s − 0.0681·7-s + 2.37·8-s + 2.64·9-s + 3.49·10-s − 0.667·11-s − 4.40·12-s − 0.123·14-s − 3.66·15-s + 2.02·16-s + 0.735·17-s + 4.80·18-s − 0.253·19-s + 4.43·20-s + 0.129·21-s − 1.21·22-s + 0.511·23-s − 4.54·24-s + 2.68·25-s − 3.12·27-s − 0.157·28-s − 0.940·29-s − 6.66·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.212369778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.212369778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 2.57T + 2T^{2} \) |
| 3 | \( 1 + 3.30T + 3T^{2} \) |
| 5 | \( 1 - 4.29T + 5T^{2} \) |
| 7 | \( 1 + 0.180T + 7T^{2} \) |
| 11 | \( 1 + 2.21T + 11T^{2} \) |
| 17 | \( 1 - 3.03T + 17T^{2} \) |
| 19 | \( 1 + 1.10T + 19T^{2} \) |
| 23 | \( 1 - 2.45T + 23T^{2} \) |
| 29 | \( 1 + 5.06T + 29T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 - 5.45T + 41T^{2} \) |
| 43 | \( 1 - 7.76T + 43T^{2} \) |
| 47 | \( 1 + 0.525T + 47T^{2} \) |
| 53 | \( 1 - 4.05T + 53T^{2} \) |
| 59 | \( 1 - 4.54T + 59T^{2} \) |
| 61 | \( 1 - 3.40T + 61T^{2} \) |
| 67 | \( 1 + 8.21T + 67T^{2} \) |
| 71 | \( 1 + 3.49T + 71T^{2} \) |
| 73 | \( 1 - 1.54T + 73T^{2} \) |
| 79 | \( 1 + 3.10T + 79T^{2} \) |
| 83 | \( 1 + 5.88T + 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 - 7.93T + 97T^{2} \) |
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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
−7.58141382743190413218470862706, −6.93052408595014084337639996451, −6.21430635770794009145558807761, −5.84250708275529133891488438399, −5.42186335931286024394193941773, −4.88211308798119039074475039592, −4.14596044042336959079029978059, −2.87926304406713189672205719172, −2.02160650716422703075699071217, −1.10996718150326526856827644136,
1.10996718150326526856827644136, 2.02160650716422703075699071217, 2.87926304406713189672205719172, 4.14596044042336959079029978059, 4.88211308798119039074475039592, 5.42186335931286024394193941773, 5.84250708275529133891488438399, 6.21430635770794009145558807761, 6.93052408595014084337639996451, 7.58141382743190413218470862706
