| L(s) = 1 | + 3.11·3-s − 5-s + 7-s + 6.72·9-s − 13-s − 3.11·15-s + 3.72·17-s + 1.72·19-s + 3.11·21-s + 1.21·23-s + 25-s + 11.6·27-s − 6.33·29-s + 7.11·31-s − 35-s + 3.90·37-s − 3.11·39-s + 3.72·41-s − 10.2·43-s − 6.72·45-s + 2.78·47-s + 49-s + 11.6·51-s + 2·53-s + 5.39·57-s + 13.3·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 1.80·3-s − 0.447·5-s + 0.377·7-s + 2.24·9-s − 0.277·13-s − 0.805·15-s + 0.904·17-s + 0.396·19-s + 0.680·21-s + 0.254·23-s + 0.200·25-s + 2.23·27-s − 1.17·29-s + 1.27·31-s − 0.169·35-s + 0.641·37-s − 0.499·39-s + 0.582·41-s − 1.56·43-s − 1.00·45-s + 0.405·47-s + 0.142·49-s + 1.62·51-s + 0.274·53-s + 0.714·57-s + 1.73·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.879591493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.879591493\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 3.11T + 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 3.72T + 17T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 23 | \( 1 - 1.21T + 23T^{2} \) |
| 29 | \( 1 + 6.33T + 29T^{2} \) |
| 31 | \( 1 - 7.11T + 31T^{2} \) |
| 37 | \( 1 - 3.90T + 37T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 2.78T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 3.29T + 67T^{2} \) |
| 71 | \( 1 + 6.23T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 1.72T + 79T^{2} \) |
| 83 | \( 1 + 1.21T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 5.80T + 97T^{2} \) |
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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
−8.472217050929470834781616696482, −7.80162321910971348930938607871, −7.49817768243142560319627084004, −6.59303621552810476118701737877, −5.36250824826443544376028398660, −4.47353853358568203516691275983, −3.70220862204438419341270941492, −3.03416869963661629802707739462, −2.20576192420505254849829142256, −1.14850584815261554197436153461,
1.14850584815261554197436153461, 2.20576192420505254849829142256, 3.03416869963661629802707739462, 3.70220862204438419341270941492, 4.47353853358568203516691275983, 5.36250824826443544376028398660, 6.59303621552810476118701737877, 7.49817768243142560319627084004, 7.80162321910971348930938607871, 8.472217050929470834781616696482
