| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s + 7-s − 8-s + 9-s − 4·11-s − 2·12-s − 14-s + 16-s − 6·17-s − 18-s − 6·19-s − 2·21-s + 4·22-s + 23-s + 2·24-s − 5·25-s + 4·27-s + 28-s − 10·29-s − 32-s + 8·33-s + 6·34-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.37·19-s − 0.436·21-s + 0.852·22-s + 0.208·23-s + 0.408·24-s − 25-s + 0.769·27-s + 0.188·28-s − 1.85·29-s − 0.176·32-s + 1.39·33-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309442 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309442 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.06220115995480, −12.53564704761836, −12.05314503406002, −11.64202930718564, −11.06608963907311, −10.90705233757546, −10.51111908157108, −10.14451602402969, −9.412741726519890, −8.921884538026604, −8.614494774776864, −8.000201825848246, −7.527011800509176, −7.160695185859138, −6.407678987955483, −6.239293643385390, −5.585666477121215, −5.213087455500775, −4.712207247288884, −4.076664573597762, −3.602563969457357, −2.560032873017225, −2.266748952090286, −1.773267810966299, −0.8438522699074406, 0, 0,
0.8438522699074406, 1.773267810966299, 2.266748952090286, 2.560032873017225, 3.602563969457357, 4.076664573597762, 4.712207247288884, 5.213087455500775, 5.585666477121215, 6.239293643385390, 6.407678987955483, 7.160695185859138, 7.527011800509176, 8.000201825848246, 8.614494774776864, 8.921884538026604, 9.412741726519890, 10.14451602402969, 10.51111908157108, 10.90705233757546, 11.06608963907311, 11.64202930718564, 12.05314503406002, 12.53564704761836, 13.06220115995480
