| L(s) = 1 | − 15.4·5-s − 23.8·7-s − 14.2·11-s + 13.8·13-s + 80.5·17-s − 144.·19-s − 141.·23-s + 112.·25-s − 251.·29-s + 16.6·31-s + 367.·35-s − 305.·37-s − 429.·41-s − 181.·43-s − 79.4·47-s + 225.·49-s − 663.·53-s + 219.·55-s − 220.·59-s + 473.·61-s − 213.·65-s − 647.·67-s + 14.4·71-s + 776.·73-s + 339.·77-s + 257.·79-s − 1.28e3·83-s + ⋯ |
| L(s) = 1 | − 1.37·5-s − 1.28·7-s − 0.390·11-s + 0.295·13-s + 1.14·17-s − 1.74·19-s − 1.27·23-s + 0.901·25-s − 1.60·29-s + 0.0965·31-s + 1.77·35-s − 1.35·37-s − 1.63·41-s − 0.644·43-s − 0.246·47-s + 0.655·49-s − 1.72·53-s + 0.538·55-s − 0.486·59-s + 0.993·61-s − 0.406·65-s − 1.18·67-s + 0.0242·71-s + 1.24·73-s + 0.502·77-s + 0.367·79-s − 1.69·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1186181059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1186181059\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 15.4T + 125T^{2} \) |
| 7 | \( 1 + 23.8T + 343T^{2} \) |
| 11 | \( 1 + 14.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 13.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 80.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 141.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 251.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 16.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 305.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 429.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 181.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 79.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 663.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 220.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 473.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 647.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 14.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 776.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 257.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 156.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.16e3T + 9.12e5T^{2} \) |
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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
−8.797001251532814768053227947356, −8.146562049973368067210063741889, −7.45288748471507346510946204860, −6.59264484946358436920412706658, −5.86420357352723047894603606616, −4.71110847689193430532551465012, −3.59322894492606852207852726764, −3.42665206386592590894863730564, −1.91065763311010522864532296909, −0.15447576520705160532441018235,
0.15447576520705160532441018235, 1.91065763311010522864532296909, 3.42665206386592590894863730564, 3.59322894492606852207852726764, 4.71110847689193430532551465012, 5.86420357352723047894603606616, 6.59264484946358436920412706658, 7.45288748471507346510946204860, 8.146562049973368067210063741889, 8.797001251532814768053227947356
