L(s) = 1 | + 1.36·2-s − 0.141·4-s − 2.50·7-s − 2.91·8-s − 11-s − 1.14·13-s − 3.41·14-s − 3.69·16-s + 7.64·17-s + 1.77·19-s − 1.36·22-s + 1.41·23-s − 1.55·26-s + 0.353·28-s + 0.726·29-s + 2.85·31-s + 0.797·32-s + 10.4·34-s + 8.42·37-s + 2.42·38-s − 0.636·41-s + 12.6·43-s + 0.141·44-s + 1.92·46-s + 6.14·47-s − 0.726·49-s + 0.161·52-s + ⋯ |
L(s) = 1 | + 0.964·2-s − 0.0706·4-s − 0.946·7-s − 1.03·8-s − 0.301·11-s − 0.316·13-s − 0.912·14-s − 0.924·16-s + 1.85·17-s + 0.407·19-s − 0.290·22-s + 0.294·23-s − 0.305·26-s + 0.0668·28-s + 0.134·29-s + 0.513·31-s + 0.141·32-s + 1.78·34-s + 1.38·37-s + 0.393·38-s − 0.0994·41-s + 1.93·43-s + 0.0213·44-s + 0.284·46-s + 0.895·47-s − 0.103·49-s + 0.0223·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.134119868\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.134119868\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 13 | \( 1 + 1.14T + 13T^{2} \) |
| 17 | \( 1 - 7.64T + 17T^{2} \) |
| 19 | \( 1 - 1.77T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 0.726T + 29T^{2} \) |
| 31 | \( 1 - 2.85T + 31T^{2} \) |
| 37 | \( 1 - 8.42T + 37T^{2} \) |
| 41 | \( 1 + 0.636T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 - 6.14T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 3.41T + 59T^{2} \) |
| 61 | \( 1 - 4.59T + 61T^{2} \) |
| 67 | \( 1 - 9.32T + 67T^{2} \) |
| 71 | \( 1 + 5.85T + 71T^{2} \) |
| 73 | \( 1 + 7.55T + 73T^{2} \) |
| 79 | \( 1 - 6.91T + 79T^{2} \) |
| 83 | \( 1 + 6.17T + 83T^{2} \) |
| 89 | \( 1 + 3.45T + 89T^{2} \) |
| 97 | \( 1 - 19.4T + 97T^{2} \) |
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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
−9.105506540971242047695313574524, −8.035083184425074321122808585108, −7.34674918441487602633326640143, −6.29477477082990396793266054897, −5.75033843238714488198867703511, −5.01094767125869889413754071595, −4.09197974434658888116327167296, −3.24983308252663908230571747431, −2.66355995427084745308497685213, −0.801515972272994065579618091862,
0.801515972272994065579618091862, 2.66355995427084745308497685213, 3.24983308252663908230571747431, 4.09197974434658888116327167296, 5.01094767125869889413754071595, 5.75033843238714488198867703511, 6.29477477082990396793266054897, 7.34674918441487602633326640143, 8.035083184425074321122808585108, 9.105506540971242047695313574524