| L(s) = 1 | + 3-s − 2·5-s − 4·7-s + 9-s − 6·13-s − 2·15-s − 6·17-s + 8·19-s − 4·21-s − 25-s + 27-s + 6·29-s + 8·35-s + 6·37-s − 6·39-s + 10·41-s + 8·43-s − 2·45-s + 9·49-s − 6·51-s + 6·53-s + 8·57-s + 4·59-s + 2·61-s − 4·63-s + 12·65-s − 12·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 1.66·13-s − 0.516·15-s − 1.45·17-s + 1.83·19-s − 0.872·21-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.35·35-s + 0.986·37-s − 0.960·39-s + 1.56·41-s + 1.21·43-s − 0.298·45-s + 9/7·49-s − 0.840·51-s + 0.824·53-s + 1.05·57-s + 0.520·59-s + 0.256·61-s − 0.503·63-s + 1.48·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.117683052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.117683052\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.932599826422474055858297718783, −7.80656913405202518952637517034, −7.35980180157774345058038770192, −6.72765071998746323530883673586, −5.76457519615442836224786884047, −4.62724126899914884066752488672, −3.98844483511472507669367285509, −2.96368714582608103142571700827, −2.50899017411562670100196244321, −0.59848239462465608494650958153,
0.59848239462465608494650958153, 2.50899017411562670100196244321, 2.96368714582608103142571700827, 3.98844483511472507669367285509, 4.62724126899914884066752488672, 5.76457519615442836224786884047, 6.72765071998746323530883673586, 7.35980180157774345058038770192, 7.80656913405202518952637517034, 8.932599826422474055858297718783
