| L(s) = 1 | + 2-s + 0.0958·3-s + 4-s − 5-s + 0.0958·6-s − 1.44·7-s + 8-s − 2.99·9-s − 10-s − 11-s + 0.0958·12-s + 1.32·13-s − 1.44·14-s − 0.0958·15-s + 16-s + 3.53·17-s − 2.99·18-s + 2.56·19-s − 20-s − 0.138·21-s − 22-s + 2.07·23-s + 0.0958·24-s + 25-s + 1.32·26-s − 0.574·27-s − 1.44·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0553·3-s + 0.5·4-s − 0.447·5-s + 0.0391·6-s − 0.544·7-s + 0.353·8-s − 0.996·9-s − 0.316·10-s − 0.301·11-s + 0.0276·12-s + 0.367·13-s − 0.384·14-s − 0.0247·15-s + 0.250·16-s + 0.856·17-s − 0.704·18-s + 0.588·19-s − 0.223·20-s − 0.0301·21-s − 0.213·22-s + 0.433·23-s + 0.0195·24-s + 0.200·25-s + 0.259·26-s − 0.110·27-s − 0.272·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 - 0.0958T + 3T^{2} \) |
| 7 | \( 1 + 1.44T + 7T^{2} \) |
| 13 | \( 1 - 1.32T + 13T^{2} \) |
| 17 | \( 1 - 3.53T + 17T^{2} \) |
| 19 | \( 1 - 2.56T + 19T^{2} \) |
| 23 | \( 1 - 2.07T + 23T^{2} \) |
| 29 | \( 1 - 4.30T + 29T^{2} \) |
| 31 | \( 1 + 7.80T + 31T^{2} \) |
| 37 | \( 1 - 7.23T + 37T^{2} \) |
| 41 | \( 1 + 5.45T + 41T^{2} \) |
| 43 | \( 1 + 8.42T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 3.14T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 3.84T + 61T^{2} \) |
| 67 | \( 1 + 3.40T + 67T^{2} \) |
| 71 | \( 1 - 9.28T + 71T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 8.91T + 83T^{2} \) |
| 89 | \( 1 + 0.475T + 89T^{2} \) |
| 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
−7.48757241412817165050650118161, −6.61568032337626621588513170211, −6.11101350756230446175224401549, −5.26265809687123574737490051545, −4.85195150622993239224102573027, −3.58927995744118953819017444485, −3.33996595170854847456229870901, −2.55612828365014578124512858639, −1.32878185460723703224528442285, 0,
1.32878185460723703224528442285, 2.55612828365014578124512858639, 3.33996595170854847456229870901, 3.58927995744118953819017444485, 4.85195150622993239224102573027, 5.26265809687123574737490051545, 6.11101350756230446175224401549, 6.61568032337626621588513170211, 7.48757241412817165050650118161
