L(s) = 1 | + 2-s + 3.25·3-s + 4-s + 3.25·6-s − 0.0778·7-s + 8-s + 7.58·9-s − 4.50·11-s + 3.25·12-s − 5.33·13-s − 0.0778·14-s + 16-s + 7.33·17-s + 7.58·18-s + 19-s − 0.253·21-s − 4.50·22-s + 3.40·23-s + 3.25·24-s − 5.33·26-s + 14.9·27-s − 0.0778·28-s − 1.33·29-s − 2.50·31-s + 32-s − 14.6·33-s + 7.33·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.87·3-s + 0.5·4-s + 1.32·6-s − 0.0294·7-s + 0.353·8-s + 2.52·9-s − 1.35·11-s + 0.939·12-s − 1.47·13-s − 0.0208·14-s + 0.250·16-s + 1.77·17-s + 1.78·18-s + 0.229·19-s − 0.0553·21-s − 0.960·22-s + 0.710·23-s + 0.664·24-s − 1.04·26-s + 2.87·27-s − 0.0147·28-s − 0.247·29-s − 0.450·31-s + 0.176·32-s − 2.55·33-s + 1.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.283853217\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.283853217\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 3.25T + 3T^{2} \) |
| 7 | \( 1 + 0.0778T + 7T^{2} \) |
| 11 | \( 1 + 4.50T + 11T^{2} \) |
| 13 | \( 1 + 5.33T + 13T^{2} \) |
| 17 | \( 1 - 7.33T + 17T^{2} \) |
| 23 | \( 1 - 3.40T + 23T^{2} \) |
| 29 | \( 1 + 1.33T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 + 5.50T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 0.506T + 43T^{2} \) |
| 47 | \( 1 + 5.66T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 7.56T + 59T^{2} \) |
| 61 | \( 1 + 2.15T + 61T^{2} \) |
| 67 | \( 1 + 4.58T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 5.09T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 7.67T + 97T^{2} \) |
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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
−9.932423726613245696562613802216, −9.284394690746707841738460036270, −8.057421319368204439048348922864, −7.69227950036236084393324339981, −6.98333121191809933373799131855, −5.38394459315033954036635176639, −4.66249845069668929723597799206, −3.31677396625629183760648040160, −2.91352497355473632961713277846, −1.82307224634606915650469816414,
1.82307224634606915650469816414, 2.91352497355473632961713277846, 3.31677396625629183760648040160, 4.66249845069668929723597799206, 5.38394459315033954036635176639, 6.98333121191809933373799131855, 7.69227950036236084393324339981, 8.057421319368204439048348922864, 9.284394690746707841738460036270, 9.932423726613245696562613802216