L(s) = 1 | + 4.24·5-s + 7.07·13-s + 2·17-s + 12.9·25-s + 9.89·29-s + 9.89·37-s − 10·41-s − 7·49-s + 7.07·53-s − 15.5·61-s + 30·65-s − 6·73-s + 8.48·85-s − 16·89-s + 8·97-s + 15.5·101-s − 18.3·109-s − 16·113-s + ⋯ |
L(s) = 1 | + 1.89·5-s + 1.96·13-s + 0.485·17-s + 2.59·25-s + 1.83·29-s + 1.62·37-s − 1.56·41-s − 49-s + 0.971·53-s − 1.99·61-s + 3.72·65-s − 0.702·73-s + 0.920·85-s − 1.69·89-s + 0.812·97-s + 1.54·101-s − 1.76·109-s − 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.956726737\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.956726737\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 4.24T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 7.07T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.89T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−7.82024147441232354302977042785, −6.62101500750266951371994805448, −6.37371137874619385016832691004, −5.78547710420675224907507661625, −5.13225893111048111711013246606, −4.30238449040890226881375698090, −3.24471990655847858004905661191, −2.62895930434644327368627332186, −1.55500791832005769255745596750, −1.10743617964424427387283479431,
1.10743617964424427387283479431, 1.55500791832005769255745596750, 2.62895930434644327368627332186, 3.24471990655847858004905661191, 4.30238449040890226881375698090, 5.13225893111048111711013246606, 5.78547710420675224907507661625, 6.37371137874619385016832691004, 6.62101500750266951371994805448, 7.82024147441232354302977042785