L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s − 15-s + 16-s − 17-s − 18-s − 19-s − 20-s + 21-s − 22-s − 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s − 15-s + 16-s − 17-s − 18-s − 19-s − 20-s + 21-s − 22-s − 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6689893536\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6689893536\) |
\(L(1)\) |
\(\approx\) |
\(0.8192921687\) |
\(L(1)\) |
\(\approx\) |
\(0.8192921687\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−35.79142811737333356269687509764, −34.70865427559987758978757840812, −33.441845568315278885518082696399, −31.89519440900383492257336751959, −30.67093331605513914671641621138, −29.80880139206916967247374645595, −27.81027347453847638739588885408, −27.20136887608357681448446831303, −26.20656294679929542975209543861, −24.70647009026482608170367867482, −24.06689097261722596730650369377, −21.72312576531163658673011952965, −20.195839335065438107780235910438, −19.659009041798644834454761882, −18.39206654255714623695067157771, −16.90701614623742805162677180682, −15.32832013590347496252759909085, −14.55176534972372880116080566063, −12.251386201593963001808289259908, −10.94231783901091159525796660923, −9.19639632655018112383849263327, −8.14261777244901882627735418720, −7.08873354104220679483739839532, −4.092812623948975180114524280798, −2.05722123429787829144165200430,
2.05722123429787829144165200430, 4.092812623948975180114524280798, 7.08873354104220679483739839532, 8.14261777244901882627735418720, 9.19639632655018112383849263327, 10.94231783901091159525796660923, 12.251386201593963001808289259908, 14.55176534972372880116080566063, 15.32832013590347496252759909085, 16.90701614623742805162677180682, 18.39206654255714623695067157771, 19.659009041798644834454761882, 20.195839335065438107780235910438, 21.72312576531163658673011952965, 24.06689097261722596730650369377, 24.70647009026482608170367867482, 26.20656294679929542975209543861, 27.20136887608357681448446831303, 27.81027347453847638739588885408, 29.80880139206916967247374645595, 30.67093331605513914671641621138, 31.89519440900383492257336751959, 33.441845568315278885518082696399, 34.70865427559987758978757840812, 35.79142811737333356269687509764