| L(s) = 1 | + 2-s + 2.18·3-s + 4-s + 2.29·5-s + 2.18·6-s + 4.57·7-s + 8-s + 1.75·9-s + 2.29·10-s − 3.50·11-s + 2.18·12-s + 4.69·13-s + 4.57·14-s + 5.00·15-s + 16-s − 5.45·17-s + 1.75·18-s + 2.39·19-s + 2.29·20-s + 9.97·21-s − 3.50·22-s + 23-s + 2.18·24-s + 0.274·25-s + 4.69·26-s − 2.71·27-s + 4.57·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.25·3-s + 0.5·4-s + 1.02·5-s + 0.890·6-s + 1.72·7-s + 0.353·8-s + 0.584·9-s + 0.726·10-s − 1.05·11-s + 0.629·12-s + 1.30·13-s + 1.22·14-s + 1.29·15-s + 0.250·16-s − 1.32·17-s + 0.413·18-s + 0.549·19-s + 0.513·20-s + 2.17·21-s − 0.747·22-s + 0.208·23-s + 0.445·24-s + 0.0548·25-s + 0.921·26-s − 0.523·27-s + 0.864·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.359507399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.359507399\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 - 2.29T + 5T^{2} \) |
| 7 | \( 1 - 4.57T + 7T^{2} \) |
| 11 | \( 1 + 3.50T + 11T^{2} \) |
| 13 | \( 1 - 4.69T + 13T^{2} \) |
| 17 | \( 1 + 5.45T + 17T^{2} \) |
| 19 | \( 1 - 2.39T + 19T^{2} \) |
| 29 | \( 1 + 4.68T + 29T^{2} \) |
| 31 | \( 1 - 6.98T + 31T^{2} \) |
| 37 | \( 1 + 5.28T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 0.611T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 4.35T + 53T^{2} \) |
| 59 | \( 1 - 9.74T + 59T^{2} \) |
| 61 | \( 1 + 6.52T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 0.752T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 0.621T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 1.81T + 97T^{2} \) |
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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
−8.231751064400275268656375913224, −7.47640031537268399528806421369, −6.67654310510199651994088415288, −5.64984591498767250724735146817, −5.26663801988645621258422826222, −4.40526129031173697489921958138, −3.63009324524276457297582942988, −2.62655678354131789954127887130, −2.08177786982483681277963206769, −1.42283851009150686706477997342,
1.42283851009150686706477997342, 2.08177786982483681277963206769, 2.62655678354131789954127887130, 3.63009324524276457297582942988, 4.40526129031173697489921958138, 5.26663801988645621258422826222, 5.64984591498767250724735146817, 6.67654310510199651994088415288, 7.47640031537268399528806421369, 8.231751064400275268656375913224
