| L(s) = 1 | − 3-s + 5-s + 9-s − 2·11-s + 13-s − 15-s − 4·17-s − 19-s + 4·23-s + 25-s − 27-s − 5·31-s + 2·33-s − 5·37-s − 39-s + 2·41-s − 9·43-s + 45-s − 2·47-s + 4·51-s + 12·53-s − 2·55-s + 57-s − 8·59-s − 14·61-s + 65-s + 9·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.258·15-s − 0.970·17-s − 0.229·19-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.898·31-s + 0.348·33-s − 0.821·37-s − 0.160·39-s + 0.312·41-s − 1.37·43-s + 0.149·45-s − 0.291·47-s + 0.560·51-s + 1.64·53-s − 0.269·55-s + 0.132·57-s − 1.04·59-s − 1.79·61-s + 0.124·65-s + 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.516700871508794049731212768795, −7.46854979156031882703419284696, −6.80713494825997564765094462723, −6.07667510012756263786527795201, −5.28213114118982570821382741764, −4.65695398070656711825159198093, −3.59596784359149244148885875796, −2.50879837305435972570932557272, −1.48514477314721654824227032560, 0,
1.48514477314721654824227032560, 2.50879837305435972570932557272, 3.59596784359149244148885875796, 4.65695398070656711825159198093, 5.28213114118982570821382741764, 6.07667510012756263786527795201, 6.80713494825997564765094462723, 7.46854979156031882703419284696, 8.516700871508794049731212768795
