| L(s) = 1 | + 2-s + 4-s + 5-s + 4.60·7-s + 8-s + 10-s + 13-s + 4.60·14-s + 16-s − 2.60·17-s − 0.605·19-s + 20-s − 2.60·23-s + 25-s + 26-s + 4.60·28-s + 2.60·29-s + 2·31-s + 32-s − 2.60·34-s + 4.60·35-s − 3.21·37-s − 0.605·38-s + 40-s − 11.2·41-s − 9.21·43-s − 2.60·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.74·7-s + 0.353·8-s + 0.316·10-s + 0.277·13-s + 1.23·14-s + 0.250·16-s − 0.631·17-s − 0.138·19-s + 0.223·20-s − 0.543·23-s + 0.200·25-s + 0.196·26-s + 0.870·28-s + 0.483·29-s + 0.359·31-s + 0.176·32-s − 0.446·34-s + 0.778·35-s − 0.527·37-s − 0.0982·38-s + 0.158·40-s − 1.75·41-s − 1.40·43-s − 0.384·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.268747537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.268747537\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4.60T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 2.60T + 17T^{2} \) |
| 19 | \( 1 + 0.605T + 19T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 3.21T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 9.21T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 5.21T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 - 7.21T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 0.605T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 0.605T + 97T^{2} \) |
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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
−10.02064011236825958252377801082, −8.676378632117119322988196481365, −8.239732828695833242203395749408, −7.18959598108737303037210264075, −6.33514148648293132517163961820, −5.29193227256447707147057990824, −4.76577449484240137330763145344, −3.77612681623027226787273635338, −2.35727679879713676818536254166, −1.49448407794268301058657094252,
1.49448407794268301058657094252, 2.35727679879713676818536254166, 3.77612681623027226787273635338, 4.76577449484240137330763145344, 5.29193227256447707147057990824, 6.33514148648293132517163961820, 7.18959598108737303037210264075, 8.239732828695833242203395749408, 8.676378632117119322988196481365, 10.02064011236825958252377801082
