L(s) = 1 | + 0.0818·2-s + 3-s − 1.99·4-s + 5-s + 0.0818·6-s + 2.84·7-s − 0.326·8-s + 9-s + 0.0818·10-s + 2.90·11-s − 1.99·12-s + 13-s + 0.232·14-s + 15-s + 3.95·16-s − 6.00·17-s + 0.0818·18-s + 7.36·19-s − 1.99·20-s + 2.84·21-s + 0.237·22-s + 3.88·23-s − 0.326·24-s + 25-s + 0.0818·26-s + 27-s − 5.66·28-s + ⋯ |
L(s) = 1 | + 0.0578·2-s + 0.577·3-s − 0.996·4-s + 0.447·5-s + 0.0333·6-s + 1.07·7-s − 0.115·8-s + 0.333·9-s + 0.0258·10-s + 0.876·11-s − 0.575·12-s + 0.277·13-s + 0.0621·14-s + 0.258·15-s + 0.989·16-s − 1.45·17-s + 0.0192·18-s + 1.68·19-s − 0.445·20-s + 0.620·21-s + 0.0506·22-s + 0.811·23-s − 0.0666·24-s + 0.200·25-s + 0.0160·26-s + 0.192·27-s − 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.922731505\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.922731505\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 - 0.0818T + 2T^{2} \) |
| 7 | \( 1 - 2.84T + 7T^{2} \) |
| 11 | \( 1 - 2.90T + 11T^{2} \) |
| 17 | \( 1 + 6.00T + 17T^{2} \) |
| 19 | \( 1 - 7.36T + 19T^{2} \) |
| 23 | \( 1 - 3.88T + 23T^{2} \) |
| 29 | \( 1 - 1.96T + 29T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 + 4.16T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 8.16T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 - 4.28T + 59T^{2} \) |
| 61 | \( 1 - 7.53T + 61T^{2} \) |
| 67 | \( 1 + 7.49T + 67T^{2} \) |
| 71 | \( 1 + 1.16T + 71T^{2} \) |
| 73 | \( 1 + 8.56T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 - 7.24T + 83T^{2} \) |
| 89 | \( 1 + 0.101T + 89T^{2} \) |
| 97 | \( 1 + 6.51T + 97T^{2} \) |
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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
−8.365846599831058936016709712792, −7.39260721826200769294965281821, −6.81495527522405765723193943219, −5.75458962462027308956764806899, −5.06970722437853851541925750567, −4.46166707474531196730549347761, −3.75187174242531639943232757757, −2.84476404543232027509900000948, −1.71916391547725775990346152178, −0.957464637888881657210769085423,
0.957464637888881657210769085423, 1.71916391547725775990346152178, 2.84476404543232027509900000948, 3.75187174242531639943232757757, 4.46166707474531196730549347761, 5.06970722437853851541925750567, 5.75458962462027308956764806899, 6.81495527522405765723193943219, 7.39260721826200769294965281821, 8.365846599831058936016709712792