| L(s) = 1 | + 0.310·2-s + 2.06·3-s − 1.90·4-s − 5-s + 0.639·6-s − 2.12·7-s − 1.21·8-s + 1.25·9-s − 0.310·10-s + 5.48·11-s − 3.92·12-s + 1.65·13-s − 0.657·14-s − 2.06·15-s + 3.43·16-s − 2.84·17-s + 0.388·18-s − 2.39·19-s + 1.90·20-s − 4.37·21-s + 1.70·22-s + 1.29·23-s − 2.49·24-s + 25-s + 0.512·26-s − 3.60·27-s + 4.03·28-s + ⋯ |
| L(s) = 1 | + 0.219·2-s + 1.19·3-s − 0.951·4-s − 0.447·5-s + 0.261·6-s − 0.801·7-s − 0.428·8-s + 0.417·9-s − 0.0981·10-s + 1.65·11-s − 1.13·12-s + 0.458·13-s − 0.175·14-s − 0.532·15-s + 0.857·16-s − 0.689·17-s + 0.0915·18-s − 0.548·19-s + 0.425·20-s − 0.954·21-s + 0.362·22-s + 0.269·23-s − 0.509·24-s + 0.200·25-s + 0.100·26-s − 0.693·27-s + 0.762·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 2 | \( 1 - 0.310T + 2T^{2} \) |
| 3 | \( 1 - 2.06T + 3T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 - 5.48T + 11T^{2} \) |
| 13 | \( 1 - 1.65T + 13T^{2} \) |
| 17 | \( 1 + 2.84T + 17T^{2} \) |
| 19 | \( 1 + 2.39T + 19T^{2} \) |
| 23 | \( 1 - 1.29T + 23T^{2} \) |
| 29 | \( 1 + 0.459T + 29T^{2} \) |
| 31 | \( 1 + 7.00T + 31T^{2} \) |
| 41 | \( 1 - 7.27T + 41T^{2} \) |
| 43 | \( 1 + 4.39T + 43T^{2} \) |
| 47 | \( 1 + 0.310T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 8.29T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 1.32T + 67T^{2} \) |
| 71 | \( 1 - 0.587T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 2.83T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−7.84881620098406716573911615243, −6.84861205243941346722539911081, −6.36946323689676939830165497350, −5.43865624605242473729445311928, −4.39601420473812279571920550107, −3.74321630612391783949478340481, −3.53539040334913199755731687566, −2.50403168674367879050124198342, −1.33845958686523469217430732513, 0,
1.33845958686523469217430732513, 2.50403168674367879050124198342, 3.53539040334913199755731687566, 3.74321630612391783949478340481, 4.39601420473812279571920550107, 5.43865624605242473729445311928, 6.36946323689676939830165497350, 6.84861205243941346722539911081, 7.84881620098406716573911615243
