| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 4·11-s − 12-s − 13-s + 16-s + 5·17-s + 18-s + 6·19-s − 4·22-s + 5·23-s − 24-s − 26-s − 27-s − 3·29-s + 7·31-s + 32-s + 4·33-s + 5·34-s + 36-s − 4·37-s + 6·38-s + 39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s + 1.21·17-s + 0.235·18-s + 1.37·19-s − 0.852·22-s + 1.04·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.557·29-s + 1.25·31-s + 0.176·32-s + 0.696·33-s + 0.857·34-s + 1/6·36-s − 0.657·37-s + 0.973·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.564068288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.564068288\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.57605818449559575227087773525, −7.30521749912677804103053895928, −6.33967474128449513463155736283, −5.65013889970476962562191841048, −5.05079344085229556490999251827, −4.68568652878365664365739747204, −3.33696187948665471692584672220, −3.05270802906748335409735694811, −1.83145909127343667584767106513, −0.75375351555194391213427468495,
0.75375351555194391213427468495, 1.83145909127343667584767106513, 3.05270802906748335409735694811, 3.33696187948665471692584672220, 4.68568652878365664365739747204, 5.05079344085229556490999251827, 5.65013889970476962562191841048, 6.33967474128449513463155736283, 7.30521749912677804103053895928, 7.57605818449559575227087773525
