| L(s) = 1 | + 3-s − 2·4-s − 7-s − 2·9-s + 3·11-s − 2·12-s − 4·13-s + 4·16-s + 6·17-s + 2·19-s − 21-s + 6·23-s − 5·25-s − 5·27-s + 2·28-s − 6·29-s − 4·31-s + 3·33-s + 4·36-s + 37-s − 4·39-s − 9·41-s + 8·43-s − 6·44-s + 3·47-s + 4·48-s − 6·49-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.577·12-s − 1.10·13-s + 16-s + 1.45·17-s + 0.458·19-s − 0.218·21-s + 1.25·23-s − 25-s − 0.962·27-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.522·33-s + 2/3·36-s + 0.164·37-s − 0.640·39-s − 1.40·41-s + 1.21·43-s − 0.904·44-s + 0.437·47-s + 0.577·48-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7256810619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7256810619\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.72216150673137017128071742797, −14.80595468440372872355464099506, −14.26876170526783296623721964932, −13.04504337075354428108495487834, −11.79612239530835321782087991524, −9.819966817497618882768827794755, −9.032345851694468357832029856913, −7.59911177067371416247669368567, −5.44973416215471530225317007593, −3.50910294340479626549928321873,
3.50910294340479626549928321873, 5.44973416215471530225317007593, 7.59911177067371416247669368567, 9.032345851694468357832029856913, 9.819966817497618882768827794755, 11.79612239530835321782087991524, 13.04504337075354428108495487834, 14.26876170526783296623721964932, 14.80595468440372872355464099506, 16.72216150673137017128071742797
