L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 4.60·7-s + 8-s + 9-s + 10-s − 12-s + 4.60·14-s − 15-s + 16-s − 4.60·17-s + 18-s − 4.60·19-s + 20-s − 4.60·21-s + 1.39·23-s − 24-s + 25-s − 27-s + 4.60·28-s + 4.60·29-s − 30-s + 6·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.74·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.288·12-s + 1.23·14-s − 0.258·15-s + 0.250·16-s − 1.11·17-s + 0.235·18-s − 1.05·19-s + 0.223·20-s − 1.00·21-s + 0.290·23-s − 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.870·28-s + 0.855·29-s − 0.182·30-s + 1.07·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.585490862\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.585490862\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 4.60T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 19 | \( 1 + 4.60T + 19T^{2} \) |
| 23 | \( 1 - 1.39T + 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 9.21T + 37T^{2} \) |
| 41 | \( 1 - 3.21T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 9.21T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 3.21T + 67T^{2} \) |
| 71 | \( 1 - 9.21T + 71T^{2} \) |
| 73 | \( 1 - 1.39T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 2.78T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 1.39T + 97T^{2} \) |
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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
−8.262753585645792121805738639425, −7.35702215847240276734848131744, −6.62468915292812895836607435464, −5.97044827693851311935706144422, −5.22621264788321009800492815264, −4.46780310863851749832043307567, −4.26719006125772752772368133974, −2.67047234609566275200523638844, −1.99810522742813569376073032605, −1.01247430591557933183622913258,
1.01247430591557933183622913258, 1.99810522742813569376073032605, 2.67047234609566275200523638844, 4.26719006125772752772368133974, 4.46780310863851749832043307567, 5.22621264788321009800492815264, 5.97044827693851311935706144422, 6.62468915292812895836607435464, 7.35702215847240276734848131744, 8.262753585645792121805738639425