| L(s) = 1 | − 2-s − 1.84·3-s + 4-s + 5-s + 1.84·6-s + 0.526·7-s − 8-s + 0.399·9-s − 10-s + 11-s − 1.84·12-s + 5.89·13-s − 0.526·14-s − 1.84·15-s + 16-s − 7.20·17-s − 0.399·18-s − 7.63·19-s + 20-s − 0.971·21-s − 22-s + 3.89·23-s + 1.84·24-s + 25-s − 5.89·26-s + 4.79·27-s + 0.526·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.06·3-s + 0.5·4-s + 0.447·5-s + 0.752·6-s + 0.199·7-s − 0.353·8-s + 0.133·9-s − 0.316·10-s + 0.301·11-s − 0.532·12-s + 1.63·13-s − 0.140·14-s − 0.476·15-s + 0.250·16-s − 1.74·17-s − 0.0941·18-s − 1.75·19-s + 0.223·20-s − 0.211·21-s − 0.213·22-s + 0.811·23-s + 0.376·24-s + 0.200·25-s − 1.15·26-s + 0.922·27-s + 0.0995·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 1.84T + 3T^{2} \) |
| 7 | \( 1 - 0.526T + 7T^{2} \) |
| 13 | \( 1 - 5.89T + 13T^{2} \) |
| 17 | \( 1 + 7.20T + 17T^{2} \) |
| 19 | \( 1 + 7.63T + 19T^{2} \) |
| 23 | \( 1 - 3.89T + 23T^{2} \) |
| 29 | \( 1 - 8.04T + 29T^{2} \) |
| 31 | \( 1 - 2.80T + 31T^{2} \) |
| 37 | \( 1 + 5.62T + 37T^{2} \) |
| 41 | \( 1 + 1.80T + 41T^{2} \) |
| 47 | \( 1 + 8.47T + 47T^{2} \) |
| 53 | \( 1 + 6.63T + 53T^{2} \) |
| 59 | \( 1 + 2.91T + 59T^{2} \) |
| 61 | \( 1 + 0.160T + 61T^{2} \) |
| 67 | \( 1 + 1.49T + 67T^{2} \) |
| 71 | \( 1 - 9.81T + 71T^{2} \) |
| 73 | \( 1 + 0.652T + 73T^{2} \) |
| 79 | \( 1 + 9.25T + 79T^{2} \) |
| 83 | \( 1 - 0.697T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
| 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
−8.343609872225976709204589322319, −6.84340461282054555617412498413, −6.39154243612876742264281933505, −6.19270469744727683994071741274, −5.00099653665708971193448429773, −4.41595787356889347287177101047, −3.21604207676774996734985454210, −2.09417507691815021202547598700, −1.19161153101949589856171613130, 0,
1.19161153101949589856171613130, 2.09417507691815021202547598700, 3.21604207676774996734985454210, 4.41595787356889347287177101047, 5.00099653665708971193448429773, 6.19270469744727683994071741274, 6.39154243612876742264281933505, 6.84340461282054555617412498413, 8.343609872225976709204589322319
