| L(s) = 1 | + 2-s + 4-s + 4.61·7-s + 8-s − 0.854·11-s + 5.61·13-s + 4.61·14-s + 16-s + 5.70·17-s − 7.09·19-s − 0.854·22-s − 8.09·23-s + 5.61·26-s + 4.61·28-s + 7.70·29-s + 2.47·31-s + 32-s + 5.70·34-s + 0.618·37-s − 7.09·38-s − 2.61·41-s − 5.70·43-s − 0.854·44-s − 8.09·46-s − 6.38·47-s + 14.3·49-s + 5.61·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.74·7-s + 0.353·8-s − 0.257·11-s + 1.55·13-s + 1.23·14-s + 0.250·16-s + 1.38·17-s − 1.62·19-s − 0.182·22-s − 1.68·23-s + 1.10·26-s + 0.872·28-s + 1.43·29-s + 0.444·31-s + 0.176·32-s + 0.978·34-s + 0.101·37-s − 1.15·38-s − 0.408·41-s − 0.870·43-s − 0.128·44-s − 1.19·46-s − 0.930·47-s + 2.04·49-s + 0.779·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.668254136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.668254136\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 + 0.854T + 11T^{2} \) |
| 13 | \( 1 - 5.61T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 + 8.09T + 23T^{2} \) |
| 29 | \( 1 - 7.70T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 0.618T + 37T^{2} \) |
| 41 | \( 1 + 2.61T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 6.38T + 47T^{2} \) |
| 53 | \( 1 + 2.61T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 + 1.23T + 67T^{2} \) |
| 71 | \( 1 + 4.47T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 - 3.52T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 - 4.56T + 89T^{2} \) |
| 97 | \( 1 - 8.18T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.572628770685638321068762769734, −8.289224492167438592598045688125, −7.64960115753189742405756679346, −6.40905405653147291229129339869, −5.88537006741001705574936064424, −4.92277502892014188189932676695, −4.29606750119704735436920386148, −3.41884280646793981492018302715, −2.11193069343791543739452659899, −1.29519668835660401044479606879,
1.29519668835660401044479606879, 2.11193069343791543739452659899, 3.41884280646793981492018302715, 4.29606750119704735436920386148, 4.92277502892014188189932676695, 5.88537006741001705574936064424, 6.40905405653147291229129339869, 7.64960115753189742405756679346, 8.289224492167438592598045688125, 8.572628770685638321068762769734
