L(s) = 1 | + 5-s − 1.55·7-s + 2.15·11-s − 13-s − 0.443·17-s − 3.71·19-s − 7.86·23-s + 25-s + 5.71·29-s − 2·31-s − 1.55·35-s + 4.15·37-s + 4.15·41-s + 12.3·43-s + 12.3·47-s − 4.57·49-s + 10.1·53-s + 2.15·55-s − 4·59-s + 8.15·61-s − 65-s + 10.3·67-s − 5.26·71-s − 0.599·73-s − 3.35·77-s − 3.04·79-s − 4·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.588·7-s + 0.649·11-s − 0.277·13-s − 0.107·17-s − 0.851·19-s − 1.64·23-s + 0.200·25-s + 1.06·29-s − 0.359·31-s − 0.263·35-s + 0.683·37-s + 0.648·41-s + 1.87·43-s + 1.79·47-s − 0.654·49-s + 1.39·53-s + 0.290·55-s − 0.520·59-s + 1.04·61-s − 0.124·65-s + 1.25·67-s − 0.625·71-s − 0.0701·73-s − 0.382·77-s − 0.342·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.834161597\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.834161597\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 1.55T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 17 | \( 1 + 0.443T + 17T^{2} \) |
| 19 | \( 1 + 3.71T + 19T^{2} \) |
| 23 | \( 1 + 7.86T + 23T^{2} \) |
| 29 | \( 1 - 5.71T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 4.15T + 37T^{2} \) |
| 41 | \( 1 - 4.15T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 8.15T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 5.26T + 71T^{2} \) |
| 73 | \( 1 + 0.599T + 73T^{2} \) |
| 79 | \( 1 + 3.04T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 7.57T + 89T^{2} \) |
| 97 | \( 1 - 6.44T + 97T^{2} \) |
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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
−8.389521320233405466234643738857, −7.52987900518149832712763830472, −6.76724335423835114074441494006, −6.09683560726035764568914941064, −5.60923403371999044047012363102, −4.35993031903667738737436733543, −3.95442746746174587667913413715, −2.73642018676201282114846998551, −2.05538361295498634417417695337, −0.74121920156900976747120944559,
0.74121920156900976747120944559, 2.05538361295498634417417695337, 2.73642018676201282114846998551, 3.95442746746174587667913413715, 4.35993031903667738737436733543, 5.60923403371999044047012363102, 6.09683560726035764568914941064, 6.76724335423835114074441494006, 7.52987900518149832712763830472, 8.389521320233405466234643738857