| L(s) = 1 | − 2-s + 4-s + 2.41·5-s + 0.521·7-s − 8-s − 2.41·10-s + 4.11·11-s − 6.34·13-s − 0.521·14-s + 16-s + 1.64·17-s − 6.78·19-s + 2.41·20-s − 4.11·22-s − 6.27·23-s + 0.852·25-s + 6.34·26-s + 0.521·28-s + 0.453·29-s − 2.03·31-s − 32-s − 1.64·34-s + 1.26·35-s + 7.37·37-s + 6.78·38-s − 2.41·40-s − 0.207·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.08·5-s + 0.196·7-s − 0.353·8-s − 0.765·10-s + 1.23·11-s − 1.75·13-s − 0.139·14-s + 0.250·16-s + 0.398·17-s − 1.55·19-s + 0.540·20-s − 0.876·22-s − 1.30·23-s + 0.170·25-s + 1.24·26-s + 0.0984·28-s + 0.0842·29-s − 0.366·31-s − 0.176·32-s − 0.281·34-s + 0.213·35-s + 1.21·37-s + 1.10·38-s − 0.382·40-s − 0.0324·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 2.41T + 5T^{2} \) |
| 7 | \( 1 - 0.521T + 7T^{2} \) |
| 11 | \( 1 - 4.11T + 11T^{2} \) |
| 13 | \( 1 + 6.34T + 13T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 + 6.78T + 19T^{2} \) |
| 23 | \( 1 + 6.27T + 23T^{2} \) |
| 29 | \( 1 - 0.453T + 29T^{2} \) |
| 31 | \( 1 + 2.03T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 + 0.207T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 - 6.60T + 47T^{2} \) |
| 53 | \( 1 + 8.35T + 53T^{2} \) |
| 59 | \( 1 + 4.41T + 59T^{2} \) |
| 61 | \( 1 - 2.95T + 61T^{2} \) |
| 67 | \( 1 - 5.02T + 67T^{2} \) |
| 71 | \( 1 + 3.57T + 71T^{2} \) |
| 73 | \( 1 + 0.738T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 4.74T + 89T^{2} \) |
| 97 | \( 1 - 0.702T + 97T^{2} \) |
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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
−7.52766237200308939202654484720, −6.82758911273285019502214538291, −6.10400753466692716645040967402, −5.71671978054634073173844538630, −4.59781526314093468988235607776, −4.01597610820394399570776939331, −2.67570706814177738946742649984, −2.13287683369942122282944025039, −1.37103502058110043594581066886, 0,
1.37103502058110043594581066886, 2.13287683369942122282944025039, 2.67570706814177738946742649984, 4.01597610820394399570776939331, 4.59781526314093468988235607776, 5.71671978054634073173844538630, 6.10400753466692716645040967402, 6.82758911273285019502214538291, 7.52766237200308939202654484720
