L(s) = 1 | + 3·3-s − 3·5-s − 2·7-s + 6·9-s − 9·15-s + 6·17-s + 4·19-s − 6·21-s − 23-s + 4·25-s + 9·27-s + 8·29-s + 7·31-s + 6·35-s − 37-s − 4·41-s + 6·43-s − 18·45-s + 8·47-s − 3·49-s + 18·51-s + 2·53-s + 12·57-s + 59-s − 4·61-s − 12·63-s + 5·67-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 1.34·5-s − 0.755·7-s + 2·9-s − 2.32·15-s + 1.45·17-s + 0.917·19-s − 1.30·21-s − 0.208·23-s + 4/5·25-s + 1.73·27-s + 1.48·29-s + 1.25·31-s + 1.01·35-s − 0.164·37-s − 0.624·41-s + 0.914·43-s − 2.68·45-s + 1.16·47-s − 3/7·49-s + 2.52·51-s + 0.274·53-s + 1.58·57-s + 0.130·59-s − 0.512·61-s − 1.51·63-s + 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.564371794\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.564371794\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.979001814119218951761473279764, −8.394339798874914668244789159293, −7.64755444346078841000957674432, −7.34464298147435526282636037125, −6.21679895699676402510107032244, −4.80898200433265196470127405375, −3.87401429056701171587588607766, −3.28681541538821572850087755824, −2.68030351930047444046477575602, −1.04001479698957226761907987971,
1.04001479698957226761907987971, 2.68030351930047444046477575602, 3.28681541538821572850087755824, 3.87401429056701171587588607766, 4.80898200433265196470127405375, 6.21679895699676402510107032244, 7.34464298147435526282636037125, 7.64755444346078841000957674432, 8.394339798874914668244789159293, 8.979001814119218951761473279764