| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 0.828·7-s + 8-s + 9-s − 0.828·11-s − 12-s − 4.82·13-s − 0.828·14-s + 16-s − 0.828·17-s + 18-s + 0.828·21-s − 0.828·22-s − 8.48·23-s − 24-s − 4.82·26-s − 27-s − 0.828·28-s + 9.65·29-s + 31-s + 32-s + 0.828·33-s − 0.828·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.313·7-s + 0.353·8-s + 0.333·9-s − 0.249·11-s − 0.288·12-s − 1.33·13-s − 0.221·14-s + 0.250·16-s − 0.200·17-s + 0.235·18-s + 0.180·21-s − 0.176·22-s − 1.76·23-s − 0.204·24-s − 0.946·26-s − 0.192·27-s − 0.156·28-s + 1.79·29-s + 0.179·31-s + 0.176·32-s + 0.144·33-s − 0.142·34-s + 0.166·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.012676360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.012676360\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 9.65T + 29T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 0.343T + 53T^{2} \) |
| 59 | \( 1 - 3.17T + 59T^{2} \) |
| 61 | \( 1 - 0.828T + 61T^{2} \) |
| 67 | \( 1 + 9.17T + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 1.65T + 83T^{2} \) |
| 89 | \( 1 - 4.82T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.911434460797894476317988500026, −7.62835159834493848515095410416, −6.56597434058042978708269898052, −6.15414211557966414519980633106, −5.35139275946801679808669644510, −4.54993292028011984577198848157, −4.06599890512267348173993887374, −2.80135678343118222346320518976, −2.21702577035937338861110953209, −0.70573891307762695189039918036,
0.70573891307762695189039918036, 2.21702577035937338861110953209, 2.80135678343118222346320518976, 4.06599890512267348173993887374, 4.54993292028011984577198848157, 5.35139275946801679808669644510, 6.15414211557966414519980633106, 6.56597434058042978708269898052, 7.62835159834493848515095410416, 7.911434460797894476317988500026
