| L(s) = 1 | + 3.39·3-s + 0.215·5-s + 8.52·9-s − 11-s + 2.12·13-s + 0.732·15-s − 0.821·17-s + 2.94·19-s + 0.732·23-s − 4.95·25-s + 18.7·27-s − 4.35·29-s + 1.91·31-s − 3.39·33-s + 9.52·37-s + 7.22·39-s − 0.821·41-s + 11.2·43-s + 1.83·45-s − 12.3·47-s − 2.78·51-s + 2·53-s − 0.215·55-s + 10.0·57-s − 5.75·59-s + 8.91·61-s + 0.459·65-s + ⋯ |
| L(s) = 1 | + 1.95·3-s + 0.0965·5-s + 2.84·9-s − 0.301·11-s + 0.589·13-s + 0.189·15-s − 0.199·17-s + 0.676·19-s + 0.152·23-s − 0.990·25-s + 3.60·27-s − 0.809·29-s + 0.343·31-s − 0.590·33-s + 1.56·37-s + 1.15·39-s − 0.128·41-s + 1.71·43-s + 0.274·45-s − 1.80·47-s − 0.390·51-s + 0.274·53-s − 0.0291·55-s + 1.32·57-s − 0.748·59-s + 1.14·61-s + 0.0569·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.151395716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.151395716\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 3.39T + 3T^{2} \) |
| 5 | \( 1 - 0.215T + 5T^{2} \) |
| 13 | \( 1 - 2.12T + 13T^{2} \) |
| 17 | \( 1 + 0.821T + 17T^{2} \) |
| 19 | \( 1 - 2.94T + 19T^{2} \) |
| 23 | \( 1 - 0.732T + 23T^{2} \) |
| 29 | \( 1 + 4.35T + 29T^{2} \) |
| 31 | \( 1 - 1.91T + 31T^{2} \) |
| 37 | \( 1 - 9.52T + 37T^{2} \) |
| 41 | \( 1 + 0.821T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 5.75T + 59T^{2} \) |
| 61 | \( 1 - 8.91T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 - 7.43T + 73T^{2} \) |
| 79 | \( 1 - 5.21T + 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 + 5.08T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.923082536160461062211567300703, −7.37126900040136833712694432602, −6.61617804854445078787268319129, −5.74222450695589967604565449695, −4.72429893223320684719175287478, −3.99071851762554253455629491516, −3.42705934706585592291868768742, −2.64653885444463869068140407472, −2.00532288405801453596276934888, −1.06635800584766769227430616406,
1.06635800584766769227430616406, 2.00532288405801453596276934888, 2.64653885444463869068140407472, 3.42705934706585592291868768742, 3.99071851762554253455629491516, 4.72429893223320684719175287478, 5.74222450695589967604565449695, 6.61617804854445078787268319129, 7.37126900040136833712694432602, 7.923082536160461062211567300703
