| L(s) = 1 | + 0.765·2-s − 1.41·4-s − 2.08·5-s + 1.23·7-s − 2.61·8-s − 1.59·10-s + 1.49·11-s + 4.10·13-s + 0.944·14-s + 0.828·16-s − 6.81·19-s + 2.94·20-s + 1.14·22-s + 3.30·23-s − 0.663·25-s + 3.14·26-s − 1.74·28-s − 6.24·29-s + 10.2·31-s + 5.86·32-s − 2.57·35-s + 3.93·37-s − 5.21·38-s + 5.44·40-s − 12.1·41-s − 3.44·43-s − 2.11·44-s + ⋯ |
| L(s) = 1 | + 0.541·2-s − 0.707·4-s − 0.931·5-s + 0.466·7-s − 0.923·8-s − 0.504·10-s + 0.451·11-s + 1.13·13-s + 0.252·14-s + 0.207·16-s − 1.56·19-s + 0.658·20-s + 0.244·22-s + 0.689·23-s − 0.132·25-s + 0.616·26-s − 0.329·28-s − 1.15·29-s + 1.84·31-s + 1.03·32-s − 0.434·35-s + 0.647·37-s − 0.845·38-s + 0.860·40-s − 1.89·41-s − 0.525·43-s − 0.319·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.765T + 2T^{2} \) |
| 5 | \( 1 + 2.08T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 1.49T + 11T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 19 | \( 1 + 6.81T + 19T^{2} \) |
| 23 | \( 1 - 3.30T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 3.93T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 3.44T + 43T^{2} \) |
| 47 | \( 1 + 1.56T + 47T^{2} \) |
| 53 | \( 1 + 3.96T + 53T^{2} \) |
| 59 | \( 1 + 8.06T + 59T^{2} \) |
| 61 | \( 1 - 3.27T + 61T^{2} \) |
| 67 | \( 1 - 2.11T + 67T^{2} \) |
| 71 | \( 1 - 0.226T + 71T^{2} \) |
| 73 | \( 1 + 0.368T + 73T^{2} \) |
| 79 | \( 1 - 2.71T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 2.68T + 97T^{2} \) |
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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
−8.266156580463292002741288640068, −8.144841413113406262850550848233, −6.77956666791742043872645718427, −6.17710005959221469313645597629, −5.17090292639885471284426403493, −4.34705280840668461454426353238, −3.89514651856670337537244609695, −3.01020619851113521092609432393, −1.45122088071617349835840119787, 0,
1.45122088071617349835840119787, 3.01020619851113521092609432393, 3.89514651856670337537244609695, 4.34705280840668461454426353238, 5.17090292639885471284426403493, 6.17710005959221469313645597629, 6.77956666791742043872645718427, 8.144841413113406262850550848233, 8.266156580463292002741288640068
