L(s) = 1 | − 2.52·2-s + 4.37·4-s − 3.46·7-s − 5.98·8-s + 11-s + 8.74·14-s + 6.37·16-s + 1.58·17-s + 4·19-s − 2.52·22-s + 0.792·23-s − 15.1·28-s − 8.74·29-s + 3.37·31-s − 4.10·32-s − 4·34-s + 1.08·37-s − 10.0·38-s − 8.74·41-s − 3.46·43-s + 4.37·44-s − 2·46-s − 6.63·47-s + 4.99·49-s + 10.0·53-s + 20.7·56-s + 22.0·58-s + ⋯ |
L(s) = 1 | − 1.78·2-s + 2.18·4-s − 1.30·7-s − 2.11·8-s + 0.301·11-s + 2.33·14-s + 1.59·16-s + 0.384·17-s + 0.917·19-s − 0.538·22-s + 0.165·23-s − 2.86·28-s − 1.62·29-s + 0.605·31-s − 0.726·32-s − 0.685·34-s + 0.178·37-s − 1.63·38-s − 1.36·41-s − 0.528·43-s + 0.659·44-s − 0.294·46-s − 0.967·47-s + 0.714·49-s + 1.38·53-s + 2.77·56-s + 2.89·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5251551977\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5251551977\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 0.792T + 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 - 1.08T + 37T^{2} \) |
| 41 | \( 1 + 8.74T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 + 6.63T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 7.37T + 59T^{2} \) |
| 61 | \( 1 + 0.744T + 61T^{2} \) |
| 67 | \( 1 + 9.30T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 1.25T + 79T^{2} \) |
| 83 | \( 1 - 6.63T + 83T^{2} \) |
| 89 | \( 1 + 1.37T + 89T^{2} \) |
| 97 | \( 1 + 5.84T + 97T^{2} \) |
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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
−9.041300363325656658920356873956, −8.322844857902938431869394498225, −7.48995684864521135684159316117, −6.87880868975869327728601818930, −6.25181260067699665293515135259, −5.30020533636251128982039308231, −3.68980322691500263352187302522, −2.93875926493938798921514204350, −1.78618455404133089448265138517, −0.59545285862512696724257669441,
0.59545285862512696724257669441, 1.78618455404133089448265138517, 2.93875926493938798921514204350, 3.68980322691500263352187302522, 5.30020533636251128982039308231, 6.25181260067699665293515135259, 6.87880868975869327728601818930, 7.48995684864521135684159316117, 8.322844857902938431869394498225, 9.041300363325656658920356873956