| L(s) = 1 | + 3-s + 4·7-s + 9-s + 5·11-s − 2·17-s + 6·19-s + 4·21-s + 3·23-s + 27-s − 10·29-s − 10·31-s + 5·33-s + 37-s + 6·41-s − 8·43-s − 4·47-s + 9·49-s − 2·51-s − 8·53-s + 6·57-s − 7·61-s + 4·63-s − 8·67-s + 3·69-s − 15·71-s + 9·73-s + 20·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.50·11-s − 0.485·17-s + 1.37·19-s + 0.872·21-s + 0.625·23-s + 0.192·27-s − 1.85·29-s − 1.79·31-s + 0.870·33-s + 0.164·37-s + 0.937·41-s − 1.21·43-s − 0.583·47-s + 9/7·49-s − 0.280·51-s − 1.09·53-s + 0.794·57-s − 0.896·61-s + 0.503·63-s − 0.977·67-s + 0.361·69-s − 1.78·71-s + 1.05·73-s + 2.27·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−13.50429381854398, −12.84012733475480, −12.29854426019317, −11.77832308298358, −11.40421034577724, −10.97235390080240, −10.74736402489014, −9.716508880929327, −9.386412930562875, −9.121793450119876, −8.620611899766600, −7.947092419679016, −7.675829312531404, −7.091221931216042, −6.793157927232819, −5.907890010389199, −5.491087706867477, −4.938439545559010, −4.391273546620950, −3.903860223184692, −3.398641749248127, −2.803561385518017, −1.837736001860871, −1.617430228645869, −1.153741868812728, 0,
1.153741868812728, 1.617430228645869, 1.837736001860871, 2.803561385518017, 3.398641749248127, 3.903860223184692, 4.391273546620950, 4.938439545559010, 5.491087706867477, 5.907890010389199, 6.793157927232819, 7.091221931216042, 7.675829312531404, 7.947092419679016, 8.620611899766600, 9.121793450119876, 9.386412930562875, 9.716508880929327, 10.74736402489014, 10.97235390080240, 11.40421034577724, 11.77832308298358, 12.29854426019317, 12.84012733475480, 13.50429381854398
