| L(s) = 1 | + 2.10·2-s + 2.45·4-s − 2.98·7-s + 0.951·8-s + 5.06·11-s + 6.06·13-s − 6.29·14-s − 2.89·16-s + 2.18·17-s + 1.81·19-s + 10.6·22-s − 3.13·23-s + 12.7·26-s − 7.31·28-s − 4.11·29-s − 31-s − 8.00·32-s + 4.59·34-s + 3.08·37-s + 3.82·38-s + 1.15·41-s + 2.56·43-s + 12.4·44-s − 6.60·46-s + 3.02·47-s + 1.90·49-s + 14.8·52-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.22·4-s − 1.12·7-s + 0.336·8-s + 1.52·11-s + 1.68·13-s − 1.68·14-s − 0.723·16-s + 0.528·17-s + 0.415·19-s + 2.27·22-s − 0.652·23-s + 2.51·26-s − 1.38·28-s − 0.764·29-s − 0.179·31-s − 1.41·32-s + 0.788·34-s + 0.506·37-s + 0.620·38-s + 0.180·41-s + 0.391·43-s + 1.87·44-s − 0.974·46-s + 0.441·47-s + 0.271·49-s + 2.06·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.858540684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.858540684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.10T + 2T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 - 5.06T + 11T^{2} \) |
| 13 | \( 1 - 6.06T + 13T^{2} \) |
| 17 | \( 1 - 2.18T + 17T^{2} \) |
| 19 | \( 1 - 1.81T + 19T^{2} \) |
| 23 | \( 1 + 3.13T + 23T^{2} \) |
| 29 | \( 1 + 4.11T + 29T^{2} \) |
| 37 | \( 1 - 3.08T + 37T^{2} \) |
| 41 | \( 1 - 1.15T + 41T^{2} \) |
| 43 | \( 1 - 2.56T + 43T^{2} \) |
| 47 | \( 1 - 3.02T + 47T^{2} \) |
| 53 | \( 1 - 0.706T + 53T^{2} \) |
| 59 | \( 1 - 7.07T + 59T^{2} \) |
| 61 | \( 1 + 4.25T + 61T^{2} \) |
| 67 | \( 1 - 1.64T + 67T^{2} \) |
| 71 | \( 1 - 5.93T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 8.59T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 - 9.06T + 89T^{2} \) |
| 97 | \( 1 - 8.54T + 97T^{2} \) |
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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
−7.72217525675826357452885233734, −6.84132043003209345210475189589, −6.17593790919385805053431512538, −6.05996122990119178463254850764, −5.13828696663923190419114828672, −4.04315314641561841816952897824, −3.71016379899778722364825501820, −3.23779168391035666578381777278, −2.07607001811922509937642032559, −0.922486049758173197668826091400,
0.922486049758173197668826091400, 2.07607001811922509937642032559, 3.23779168391035666578381777278, 3.71016379899778722364825501820, 4.04315314641561841816952897824, 5.13828696663923190419114828672, 6.05996122990119178463254850764, 6.17593790919385805053431512538, 6.84132043003209345210475189589, 7.72217525675826357452885233734
