| L(s) = 1 | − 1.44·2-s − 2.35·3-s + 0.0964·4-s − 1.54·5-s + 3.41·6-s + 2.00·7-s + 2.75·8-s + 2.56·9-s + 2.23·10-s + 4.24·11-s − 0.227·12-s + 0.309·13-s − 2.89·14-s + 3.64·15-s − 4.18·16-s − 2.86·17-s − 3.70·18-s − 7.71·19-s − 0.148·20-s − 4.72·21-s − 6.15·22-s + 1.87·23-s − 6.49·24-s − 2.61·25-s − 0.448·26-s + 1.03·27-s + 0.193·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s − 1.36·3-s + 0.0482·4-s − 0.690·5-s + 1.39·6-s + 0.756·7-s + 0.974·8-s + 0.853·9-s + 0.707·10-s + 1.28·11-s − 0.0656·12-s + 0.0859·13-s − 0.774·14-s + 0.940·15-s − 1.04·16-s − 0.695·17-s − 0.873·18-s − 1.76·19-s − 0.0332·20-s − 1.03·21-s − 1.31·22-s + 0.391·23-s − 1.32·24-s − 0.522·25-s − 0.0880·26-s + 0.199·27-s + 0.0364·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 401 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 401 | \( 1 + T \) |
| good | 2 | \( 1 + 1.44T + 2T^{2} \) |
| 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 + 1.54T + 5T^{2} \) |
| 7 | \( 1 - 2.00T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 - 0.309T + 13T^{2} \) |
| 17 | \( 1 + 2.86T + 17T^{2} \) |
| 19 | \( 1 + 7.71T + 19T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 - 8.85T + 29T^{2} \) |
| 31 | \( 1 + 8.84T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 + 9.08T + 43T^{2} \) |
| 47 | \( 1 - 5.36T + 47T^{2} \) |
| 53 | \( 1 + 2.25T + 53T^{2} \) |
| 59 | \( 1 + 5.34T + 59T^{2} \) |
| 61 | \( 1 + 1.34T + 61T^{2} \) |
| 67 | \( 1 + 6.27T + 67T^{2} \) |
| 71 | \( 1 - 5.10T + 71T^{2} \) |
| 73 | \( 1 + 3.61T + 73T^{2} \) |
| 79 | \( 1 + 9.16T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−10.96751617496031372322646830820, −10.05466938015904423464812953855, −8.838886779419103484883504398319, −8.268121876066490875409867893021, −7.04425833604720666953751000249, −6.26715263589075605979265832265, −4.80411971188846825452435950301, −4.16955528548761491998370361645, −1.52536752754547509875888279223, 0,
1.52536752754547509875888279223, 4.16955528548761491998370361645, 4.80411971188846825452435950301, 6.26715263589075605979265832265, 7.04425833604720666953751000249, 8.268121876066490875409867893021, 8.838886779419103484883504398319, 10.05466938015904423464812953855, 10.96751617496031372322646830820
