| L(s) = 1 | − 0.509·2-s + 3-s − 1.74·4-s − 4.41·5-s − 0.509·6-s + 1.90·8-s + 9-s + 2.24·10-s − 1.67·11-s − 1.74·12-s + 4.66·13-s − 4.41·15-s + 2.50·16-s − 6.24·17-s − 0.509·18-s + 0.694·19-s + 7.68·20-s + 0.853·22-s − 23-s + 1.90·24-s + 14.4·25-s − 2.37·26-s + 27-s + 5.60·29-s + 2.24·30-s − 4.24·31-s − 5.09·32-s + ⋯ |
| L(s) = 1 | − 0.360·2-s + 0.577·3-s − 0.870·4-s − 1.97·5-s − 0.208·6-s + 0.673·8-s + 0.333·9-s + 0.711·10-s − 0.505·11-s − 0.502·12-s + 1.29·13-s − 1.14·15-s + 0.627·16-s − 1.51·17-s − 0.120·18-s + 0.159·19-s + 1.71·20-s + 0.181·22-s − 0.208·23-s + 0.389·24-s + 2.89·25-s − 0.466·26-s + 0.192·27-s + 1.03·29-s + 0.410·30-s − 0.763·31-s − 0.899·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.509T + 2T^{2} \) |
| 5 | \( 1 + 4.41T + 5T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 - 4.66T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 - 0.694T + 19T^{2} \) |
| 29 | \( 1 - 5.60T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 - 9.26T + 37T^{2} \) |
| 41 | \( 1 - 5.15T + 41T^{2} \) |
| 43 | \( 1 - 4.20T + 43T^{2} \) |
| 47 | \( 1 - 1.92T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 59 | \( 1 + 9.39T + 59T^{2} \) |
| 61 | \( 1 + 7.72T + 61T^{2} \) |
| 67 | \( 1 + 8.22T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 0.581T + 79T^{2} \) |
| 83 | \( 1 + 6.95T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.135411228148651268676879885519, −7.87672701659035158453201684775, −7.09976276293408730403416242056, −6.08667735538020527543551096803, −4.72296058376031889956737811401, −4.29572512991802543720430364627, −3.66877486657852383253170447251, −2.76838653666388448197415794682, −1.10994792929480619463741861328, 0,
1.10994792929480619463741861328, 2.76838653666388448197415794682, 3.66877486657852383253170447251, 4.29572512991802543720430364627, 4.72296058376031889956737811401, 6.08667735538020527543551096803, 7.09976276293408730403416242056, 7.87672701659035158453201684775, 8.135411228148651268676879885519
