| L(s) = 1 | + 0.322·3-s − 2.48·5-s − 0.274·7-s − 2.89·9-s + 2.55·11-s − 3.41·13-s − 0.802·15-s − 3.52·17-s + 3.46·19-s − 0.0886·21-s + 1.18·25-s − 1.90·27-s − 1.16·29-s + 8.51·31-s + 0.824·33-s + 0.683·35-s + 0.463·37-s − 1.10·39-s − 11.4·41-s − 3.94·43-s + 7.20·45-s + 0.258·47-s − 6.92·49-s − 1.13·51-s − 1.04·53-s − 6.36·55-s + 1.11·57-s + ⋯ |
| L(s) = 1 | + 0.186·3-s − 1.11·5-s − 0.103·7-s − 0.965·9-s + 0.770·11-s − 0.946·13-s − 0.207·15-s − 0.855·17-s + 0.794·19-s − 0.0193·21-s + 0.237·25-s − 0.365·27-s − 0.215·29-s + 1.52·31-s + 0.143·33-s + 0.115·35-s + 0.0762·37-s − 0.176·39-s − 1.79·41-s − 0.602·43-s + 1.07·45-s + 0.0377·47-s − 0.989·49-s − 0.159·51-s − 0.144·53-s − 0.857·55-s + 0.147·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.028597380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.028597380\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 0.322T + 3T^{2} \) |
| 5 | \( 1 + 2.48T + 5T^{2} \) |
| 7 | \( 1 + 0.274T + 7T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 + 3.52T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 29 | \( 1 + 1.16T + 29T^{2} \) |
| 31 | \( 1 - 8.51T + 31T^{2} \) |
| 37 | \( 1 - 0.463T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 3.94T + 43T^{2} \) |
| 47 | \( 1 - 0.258T + 47T^{2} \) |
| 53 | \( 1 + 1.04T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 5.79T + 61T^{2} \) |
| 67 | \( 1 - 3.94T + 67T^{2} \) |
| 71 | \( 1 + 6.19T + 71T^{2} \) |
| 73 | \( 1 + 4.03T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 7.96T + 89T^{2} \) |
| 97 | \( 1 - 7.27T + 97T^{2} \) |
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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
−8.277101619610594853574058970608, −7.84167831338664538544537199654, −6.91149235582734073427368686073, −6.41416513400990038779127437822, −5.27504413543049746137363741844, −4.63291003811895204505139684836, −3.70764302590033508763969277599, −3.08697936317218016481048963661, −2.07108490188315220398190585019, −0.54620442237640254295606961864,
0.54620442237640254295606961864, 2.07108490188315220398190585019, 3.08697936317218016481048963661, 3.70764302590033508763969277599, 4.63291003811895204505139684836, 5.27504413543049746137363741844, 6.41416513400990038779127437822, 6.91149235582734073427368686073, 7.84167831338664538544537199654, 8.277101619610594853574058970608
