| L(s) = 1 | − 2·5-s + 2·11-s + 3·13-s + 8·17-s − 19-s + 8·23-s − 25-s − 4·29-s + 3·31-s − 37-s + 6·41-s − 11·43-s − 6·47-s + 12·53-s − 4·55-s − 4·59-s + 6·61-s − 6·65-s − 13·67-s − 10·71-s + 11·73-s + 3·79-s − 2·83-s − 16·85-s + 2·95-s − 10·97-s + 10·101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.603·11-s + 0.832·13-s + 1.94·17-s − 0.229·19-s + 1.66·23-s − 1/5·25-s − 0.742·29-s + 0.538·31-s − 0.164·37-s + 0.937·41-s − 1.67·43-s − 0.875·47-s + 1.64·53-s − 0.539·55-s − 0.520·59-s + 0.768·61-s − 0.744·65-s − 1.58·67-s − 1.18·71-s + 1.28·73-s + 0.337·79-s − 0.219·83-s − 1.73·85-s + 0.205·95-s − 1.01·97-s + 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.962042379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.962042379\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.902948231459067302935552346558, −7.33475219528809330872600366668, −6.62591164060096215810738704248, −5.81204583549235331357934123304, −5.13277720066640461862378462543, −4.21939064268804023339027371565, −3.53018349026108514725570576258, −3.02958828511922128937801832292, −1.59275250042016111667743471768, −0.76499152982008805008093192117,
0.76499152982008805008093192117, 1.59275250042016111667743471768, 3.02958828511922128937801832292, 3.53018349026108514725570576258, 4.21939064268804023339027371565, 5.13277720066640461862378462543, 5.81204583549235331357934123304, 6.62591164060096215810738704248, 7.33475219528809330872600366668, 7.902948231459067302935552346558
