L(s) = 1 | − 0.628·2-s − 3-s − 1.60·4-s + 5-s + 0.628·6-s + 4.14·7-s + 2.26·8-s + 9-s − 0.628·10-s + 1.60·12-s + 5.40·13-s − 2.60·14-s − 15-s + 1.78·16-s + 4.14·17-s − 0.628·18-s − 1.25·19-s − 1.60·20-s − 4.14·21-s + 5.21·23-s − 2.26·24-s + 25-s − 3.39·26-s − 27-s − 6.66·28-s − 7.04·29-s + 0.628·30-s + ⋯ |
L(s) = 1 | − 0.444·2-s − 0.577·3-s − 0.802·4-s + 0.447·5-s + 0.256·6-s + 1.56·7-s + 0.800·8-s + 0.333·9-s − 0.198·10-s + 0.463·12-s + 1.49·13-s − 0.696·14-s − 0.258·15-s + 0.447·16-s + 1.00·17-s − 0.148·18-s − 0.288·19-s − 0.359·20-s − 0.905·21-s + 1.08·23-s − 0.462·24-s + 0.200·25-s − 0.665·26-s − 0.192·27-s − 1.25·28-s − 1.30·29-s + 0.114·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.404660722\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.404660722\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 0.628T + 2T^{2} \) |
| 7 | \( 1 - 4.14T + 7T^{2} \) |
| 13 | \( 1 - 5.40T + 13T^{2} \) |
| 17 | \( 1 - 4.14T + 17T^{2} \) |
| 19 | \( 1 + 1.25T + 19T^{2} \) |
| 23 | \( 1 - 5.21T + 23T^{2} \) |
| 29 | \( 1 + 7.04T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 7.21T + 37T^{2} \) |
| 41 | \( 1 + 9.55T + 41T^{2} \) |
| 43 | \( 1 - 6.66T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 5.21T + 59T^{2} \) |
| 61 | \( 1 - 8.29T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 5.21T + 71T^{2} \) |
| 73 | \( 1 + 2.89T + 73T^{2} \) |
| 79 | \( 1 + 1.25T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 19.2T + 97T^{2} \) |
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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
−9.057272049165713441974478275076, −8.598568608138164357181224917399, −7.86435221270655554681965915800, −7.03486740796409382403463680434, −5.77856292238863530929805054913, −5.28754219451573072544295766581, −4.47847650461691872698979398435, −3.54141246293486544080317345458, −1.71896941413212670191744613550, −1.01733217097911136239513203341,
1.01733217097911136239513203341, 1.71896941413212670191744613550, 3.54141246293486544080317345458, 4.47847650461691872698979398435, 5.28754219451573072544295766581, 5.77856292238863530929805054913, 7.03486740796409382403463680434, 7.86435221270655554681965915800, 8.598568608138164357181224917399, 9.057272049165713441974478275076