| L(s) = 1 | − 1.83·2-s + 1.36·4-s + 0.635·5-s + 1.16·8-s − 1.16·10-s + 11-s − 1.80·13-s − 4.86·16-s + 2.83·17-s − 5.56·19-s + 0.867·20-s − 1.83·22-s − 2.16·23-s − 4.59·25-s + 3.30·26-s + 10.4·29-s + 6.43·31-s + 6.59·32-s − 5.19·34-s + 6.06·37-s + 10.2·38-s + 0.740·40-s − 7.53·41-s − 4.86·43-s + 1.36·44-s + 3.97·46-s + 2.83·47-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.682·4-s + 0.284·5-s + 0.412·8-s − 0.368·10-s + 0.301·11-s − 0.499·13-s − 1.21·16-s + 0.687·17-s − 1.27·19-s + 0.193·20-s − 0.391·22-s − 0.451·23-s − 0.919·25-s + 0.647·26-s + 1.93·29-s + 1.15·31-s + 1.16·32-s − 0.891·34-s + 0.997·37-s + 1.65·38-s + 0.117·40-s − 1.17·41-s − 0.742·43-s + 0.205·44-s + 0.585·46-s + 0.413·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 5 | \( 1 - 0.635T + 5T^{2} \) |
| 13 | \( 1 + 1.80T + 13T^{2} \) |
| 17 | \( 1 - 2.83T + 17T^{2} \) |
| 19 | \( 1 + 5.56T + 19T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 6.43T + 31T^{2} \) |
| 37 | \( 1 - 6.06T + 37T^{2} \) |
| 41 | \( 1 + 7.53T + 41T^{2} \) |
| 43 | \( 1 + 4.86T + 43T^{2} \) |
| 47 | \( 1 - 2.83T + 47T^{2} \) |
| 53 | \( 1 + 7.46T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 4.33T + 61T^{2} \) |
| 67 | \( 1 + 1.60T + 67T^{2} \) |
| 71 | \( 1 + 4.29T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 - 4.76T + 79T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 + 0.364T + 89T^{2} \) |
| 97 | \( 1 + 2.59T + 97T^{2} \) |
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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
−8.106165609700729675105921321517, −7.45941094955120855074178501111, −6.52252303086122595500984840307, −6.09272801851864657852884227908, −4.81360420275503999313988601256, −4.32559619522906264822015630598, −3.04340704999059023297007068751, −2.08762369836644422024915241969, −1.21160148908528301889068397631, 0,
1.21160148908528301889068397631, 2.08762369836644422024915241969, 3.04340704999059023297007068751, 4.32559619522906264822015630598, 4.81360420275503999313988601256, 6.09272801851864657852884227908, 6.52252303086122595500984840307, 7.45941094955120855074178501111, 8.106165609700729675105921321517
