| L(s) = 1 | + 3·3-s + 5-s − 7-s + 6·9-s + 11-s + 4·13-s + 3·15-s + 2·17-s + 6·19-s − 3·21-s − 5·23-s − 4·25-s + 9·27-s − 10·29-s + 31-s + 3·33-s − 35-s + 5·37-s + 12·39-s − 2·41-s + 8·43-s + 6·45-s + 8·47-s + 49-s + 6·51-s + 6·53-s + 55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s − 0.377·7-s + 2·9-s + 0.301·11-s + 1.10·13-s + 0.774·15-s + 0.485·17-s + 1.37·19-s − 0.654·21-s − 1.04·23-s − 4/5·25-s + 1.73·27-s − 1.85·29-s + 0.179·31-s + 0.522·33-s − 0.169·35-s + 0.821·37-s + 1.92·39-s − 0.312·41-s + 1.21·43-s + 0.894·45-s + 1.16·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.525174198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.525174198\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.235973671576885980607443023052, −7.68970417072178479885822767810, −7.09870504941216159237813757818, −6.03724208816964924840584830133, −5.49802456090611138870760412574, −3.97360161654142797734458721205, −3.80182798373306504535013698839, −2.87038052167988176348767767638, −2.05700932969856102340137560896, −1.18040705127292800697184070233,
1.18040705127292800697184070233, 2.05700932969856102340137560896, 2.87038052167988176348767767638, 3.80182798373306504535013698839, 3.97360161654142797734458721205, 5.49802456090611138870760412574, 6.03724208816964924840584830133, 7.09870504941216159237813757818, 7.68970417072178479885822767810, 8.235973671576885980607443023052
