| L(s) = 1 | + 5-s + 1.41·7-s + 0.585·11-s + 0.585·13-s + 2.82·17-s + 19-s + 4.82·23-s + 25-s + 7.07·29-s − 4.82·31-s + 1.41·35-s + 6.24·37-s − 9.89·41-s + 11.0·43-s + 3.17·47-s − 5·49-s − 8.48·53-s + 0.585·55-s + 1.17·59-s − 1.65·61-s + 0.585·65-s − 11.3·67-s + 14.8·71-s − 11.6·73-s + 0.828·77-s − 13.6·79-s + 7.65·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.534·7-s + 0.176·11-s + 0.162·13-s + 0.685·17-s + 0.229·19-s + 1.00·23-s + 0.200·25-s + 1.31·29-s − 0.867·31-s + 0.239·35-s + 1.02·37-s − 1.54·41-s + 1.68·43-s + 0.462·47-s − 0.714·49-s − 1.16·53-s + 0.0789·55-s + 0.152·59-s − 0.212·61-s + 0.0726·65-s − 1.38·67-s + 1.75·71-s − 1.36·73-s + 0.0944·77-s − 1.53·79-s + 0.840·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.686761836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.686761836\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 0.585T + 11T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 3.17T + 47T^{2} \) |
| 53 | \( 1 + 8.48T + 53T^{2} \) |
| 59 | \( 1 - 1.17T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 7.65T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.87075414684223591097796688509, −7.37370908730404227956166364003, −6.48994960882343430136223880575, −5.89196867383542511109516819655, −5.07090093319266017194171338144, −4.54470412197915323416869552356, −3.48737900461078471150091161812, −2.76315808331837961158412007014, −1.72254258018329980200785082085, −0.893125164363242432373359258361,
0.893125164363242432373359258361, 1.72254258018329980200785082085, 2.76315808331837961158412007014, 3.48737900461078471150091161812, 4.54470412197915323416869552356, 5.07090093319266017194171338144, 5.89196867383542511109516819655, 6.48994960882343430136223880575, 7.37370908730404227956166364003, 7.87075414684223591097796688509
