| L(s) = 1 | + 0.461·2-s − 3.03·3-s − 1.78·4-s − 2.42·5-s − 1.40·6-s − 0.344·7-s − 1.74·8-s + 6.19·9-s − 1.11·10-s + 5.41·12-s + 2.06·13-s − 0.158·14-s + 7.34·15-s + 2.76·16-s + 1.43·17-s + 2.86·18-s + 4.86·19-s + 4.32·20-s + 1.04·21-s − 0.690·23-s + 5.30·24-s + 0.866·25-s + 0.954·26-s − 9.70·27-s + 0.614·28-s + 6.16·29-s + 3.39·30-s + ⋯ |
| L(s) = 1 | + 0.326·2-s − 1.75·3-s − 0.893·4-s − 1.08·5-s − 0.572·6-s − 0.130·7-s − 0.618·8-s + 2.06·9-s − 0.353·10-s + 1.56·12-s + 0.572·13-s − 0.0424·14-s + 1.89·15-s + 0.691·16-s + 0.348·17-s + 0.675·18-s + 1.11·19-s + 0.967·20-s + 0.227·21-s − 0.143·23-s + 1.08·24-s + 0.173·25-s + 0.187·26-s − 1.86·27-s + 0.116·28-s + 1.14·29-s + 0.619·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6502935518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6502935518\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.461T + 2T^{2} \) |
| 3 | \( 1 + 3.03T + 3T^{2} \) |
| 5 | \( 1 + 2.42T + 5T^{2} \) |
| 7 | \( 1 + 0.344T + 7T^{2} \) |
| 13 | \( 1 - 2.06T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 - 4.86T + 19T^{2} \) |
| 23 | \( 1 + 0.690T + 23T^{2} \) |
| 29 | \( 1 - 6.16T + 29T^{2} \) |
| 31 | \( 1 - 9.07T + 31T^{2} \) |
| 37 | \( 1 + 0.240T + 37T^{2} \) |
| 41 | \( 1 - 1.57T + 41T^{2} \) |
| 43 | \( 1 - 0.272T + 43T^{2} \) |
| 47 | \( 1 + 6.07T + 47T^{2} \) |
| 53 | \( 1 + 1.80T + 53T^{2} \) |
| 59 | \( 1 - 4.53T + 59T^{2} \) |
| 67 | \( 1 + 5.27T + 67T^{2} \) |
| 71 | \( 1 - 4.61T + 71T^{2} \) |
| 73 | \( 1 + 7.47T + 73T^{2} \) |
| 79 | \( 1 - 0.695T + 79T^{2} \) |
| 83 | \( 1 - 0.407T + 83T^{2} \) |
| 89 | \( 1 - 3.76T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−7.895619612334220359829013841861, −7.05114746695508021139298537761, −6.27023921202971289838104490714, −5.80834122834335765350583469898, −4.88039309442464974227724100427, −4.64264101280506208703666251183, −3.79412898347070092125163304709, −3.10294622479097998077376230377, −1.19863305841466323211846751225, −0.51459815528320061175958466598,
0.51459815528320061175958466598, 1.19863305841466323211846751225, 3.10294622479097998077376230377, 3.79412898347070092125163304709, 4.64264101280506208703666251183, 4.88039309442464974227724100427, 5.80834122834335765350583469898, 6.27023921202971289838104490714, 7.05114746695508021139298537761, 7.895619612334220359829013841861
