| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 11-s − 13-s + 15-s + 17-s + 19-s + 21-s − 23-s + 25-s − 27-s + 29-s − 31-s + 33-s + 35-s + 37-s + 39-s + 41-s + 43-s − 45-s + 47-s + 49-s − 51-s − 53-s + 55-s + ⋯ |
| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 11-s − 13-s + 15-s + 17-s + 19-s + 21-s − 23-s + 25-s − 27-s + 29-s − 31-s + 33-s + 35-s + 37-s + 39-s + 41-s + 43-s − 45-s + 47-s + 49-s − 51-s − 53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5026107595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5026107595\) |
| \(L(1)\) |
\(\approx\) |
\(0.5597789832\) |
| \(L(1)\) |
\(\approx\) |
\(0.5597789832\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 83 | \( 1 \) |
| good | 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 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 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 | \( 1 + T \) |
| 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
−24.83254816363774151765423875422, −23.781920676827283311362797110193, −23.33553398288746065968708883780, −22.43004588714711604641464447784, −21.79384372216510298346008737775, −20.51103488249294423365963246552, −19.53623273815445797171382400396, −18.73953552027787235234382124769, −17.92977663703181421579991686042, −16.67744282880487592404058814852, −16.071466519828879266809626342530, −15.46919406680024012346160871457, −14.1236377871988387992872032401, −12.62283739373281771556403293861, −12.38461542223362586866311152303, −11.29829501702516948747932271110, −10.27463009006876752520343022912, −9.53564479602287805090211332643, −7.8005564604420591172376017711, −7.268189094752791452220528135379, −5.994341151584676918132507561827, −5.060751843099240661420427086181, −3.926285786246927379078935982077, −2.739296607122986250007097771382, −0.66276616693936006583367575935,
0.66276616693936006583367575935, 2.739296607122986250007097771382, 3.926285786246927379078935982077, 5.060751843099240661420427086181, 5.994341151584676918132507561827, 7.268189094752791452220528135379, 7.8005564604420591172376017711, 9.53564479602287805090211332643, 10.27463009006876752520343022912, 11.29829501702516948747932271110, 12.38461542223362586866311152303, 12.62283739373281771556403293861, 14.1236377871988387992872032401, 15.46919406680024012346160871457, 16.071466519828879266809626342530, 16.67744282880487592404058814852, 17.92977663703181421579991686042, 18.73953552027787235234382124769, 19.53623273815445797171382400396, 20.51103488249294423365963246552, 21.79384372216510298346008737775, 22.43004588714711604641464447784, 23.33553398288746065968708883780, 23.781920676827283311362797110193, 24.83254816363774151765423875422
