| L(s) = 1 | − 2.48·3-s + 3.05·7-s + 3.19·9-s + 5.36·11-s + 4.59·13-s − 17-s − 4.57·19-s − 7.60·21-s + 1.24·23-s − 0.483·27-s − 5.93·29-s − 9.84·31-s − 13.3·33-s − 4.20·37-s − 11.4·39-s + 0.404·41-s − 5.76·43-s − 3.35·47-s + 2.34·49-s + 2.48·51-s − 4.81·53-s + 11.3·57-s − 12.7·59-s + 4.97·61-s + 9.76·63-s + 6.82·67-s − 3.10·69-s + ⋯ |
| L(s) = 1 | − 1.43·3-s + 1.15·7-s + 1.06·9-s + 1.61·11-s + 1.27·13-s − 0.242·17-s − 1.04·19-s − 1.66·21-s + 0.260·23-s − 0.0931·27-s − 1.10·29-s − 1.76·31-s − 2.32·33-s − 0.691·37-s − 1.83·39-s + 0.0631·41-s − 0.878·43-s − 0.489·47-s + 0.335·49-s + 0.348·51-s − 0.661·53-s + 1.50·57-s − 1.65·59-s + 0.637·61-s + 1.23·63-s + 0.834·67-s − 0.373·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2.48T + 3T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 - 5.36T + 11T^{2} \) |
| 13 | \( 1 - 4.59T + 13T^{2} \) |
| 19 | \( 1 + 4.57T + 19T^{2} \) |
| 23 | \( 1 - 1.24T + 23T^{2} \) |
| 29 | \( 1 + 5.93T + 29T^{2} \) |
| 31 | \( 1 + 9.84T + 31T^{2} \) |
| 37 | \( 1 + 4.20T + 37T^{2} \) |
| 41 | \( 1 - 0.404T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 + 3.35T + 47T^{2} \) |
| 53 | \( 1 + 4.81T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 4.97T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 - 4.11T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 2.27T + 97T^{2} \) |
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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
−7.39815895200559954216671018301, −6.77363640487882316541966783111, −6.11331433679717001518634796228, −5.64348860408099591576460733927, −4.78584396093622962313899632548, −4.18645495771147220450248191168, −3.46800198074770899206063282911, −1.70409018876705742091366568202, −1.41282187438609532032062479655, 0,
1.41282187438609532032062479655, 1.70409018876705742091366568202, 3.46800198074770899206063282911, 4.18645495771147220450248191168, 4.78584396093622962313899632548, 5.64348860408099591576460733927, 6.11331433679717001518634796228, 6.77363640487882316541966783111, 7.39815895200559954216671018301
