| L(s) = 1 | + 3-s − 1.69·5-s − 4.12·7-s + 9-s + 2.48·11-s + 1.35·13-s − 1.69·15-s − 7.89·17-s + 5.08·19-s − 4.12·21-s + 6.55·23-s − 2.13·25-s + 27-s + 7.41·29-s + 4.69·31-s + 2.48·33-s + 6.99·35-s − 0.0133·37-s + 1.35·39-s − 1.02·41-s − 8.67·43-s − 1.69·45-s + 5.31·47-s + 10.0·49-s − 7.89·51-s + 12.1·53-s − 4.21·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.757·5-s − 1.56·7-s + 0.333·9-s + 0.750·11-s + 0.375·13-s − 0.437·15-s − 1.91·17-s + 1.16·19-s − 0.901·21-s + 1.36·23-s − 0.426·25-s + 0.192·27-s + 1.37·29-s + 0.843·31-s + 0.433·33-s + 1.18·35-s − 0.00219·37-s + 0.216·39-s − 0.159·41-s − 1.32·43-s − 0.252·45-s + 0.775·47-s + 1.43·49-s − 1.10·51-s + 1.66·53-s − 0.568·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.530807682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.530807682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 1.69T + 5T^{2} \) |
| 7 | \( 1 + 4.12T + 7T^{2} \) |
| 11 | \( 1 - 2.48T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 + 7.89T + 17T^{2} \) |
| 19 | \( 1 - 5.08T + 19T^{2} \) |
| 23 | \( 1 - 6.55T + 23T^{2} \) |
| 29 | \( 1 - 7.41T + 29T^{2} \) |
| 31 | \( 1 - 4.69T + 31T^{2} \) |
| 37 | \( 1 + 0.0133T + 37T^{2} \) |
| 41 | \( 1 + 1.02T + 41T^{2} \) |
| 43 | \( 1 + 8.67T + 43T^{2} \) |
| 47 | \( 1 - 5.31T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 0.0660T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 6.10T + 67T^{2} \) |
| 71 | \( 1 + 9.95T + 71T^{2} \) |
| 73 | \( 1 + 3.76T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 6.46T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−8.950338844672650928791589107750, −8.681638055729251297915474125086, −7.50971232055187691314296533104, −6.73341563161494896300410016263, −6.37716933603144046745301276243, −4.95641239946257351618449574890, −3.97870150717929260792494782856, −3.35948881908411466769497806693, −2.49255146751191897159565127050, −0.795042541549264012198331894224,
0.795042541549264012198331894224, 2.49255146751191897159565127050, 3.35948881908411466769497806693, 3.97870150717929260792494782856, 4.95641239946257351618449574890, 6.37716933603144046745301276243, 6.73341563161494896300410016263, 7.50971232055187691314296533104, 8.681638055729251297915474125086, 8.950338844672650928791589107750
