| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 2·11-s + 4·13-s + 15-s − 2·17-s + 6·19-s + 21-s − 4·23-s + 25-s − 27-s + 10·29-s + 2·31-s + 2·33-s + 35-s + 2·37-s − 4·39-s − 10·41-s − 4·43-s − 45-s − 8·47-s + 49-s + 2·51-s − 4·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s + 0.258·15-s − 0.485·17-s + 1.37·19-s + 0.218·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.85·29-s + 0.359·31-s + 0.348·33-s + 0.169·35-s + 0.328·37-s − 0.640·39-s − 1.56·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.549·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.206722130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.206722130\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.132162141004278628820075246898, −7.17368938044202750880248761636, −6.52548649112790484851642687161, −5.97063312592487453494601153722, −5.08933456511422780425846175707, −4.51004545981651044037181349260, −3.52358036473338736182224016045, −2.94471302927486524782829615668, −1.64317107815460195313063922484, −0.59460270851446754818728977406,
0.59460270851446754818728977406, 1.64317107815460195313063922484, 2.94471302927486524782829615668, 3.52358036473338736182224016045, 4.51004545981651044037181349260, 5.08933456511422780425846175707, 5.97063312592487453494601153722, 6.52548649112790484851642687161, 7.17368938044202750880248761636, 8.132162141004278628820075246898
