| L(s) = 1 | − 2-s − 2.74·3-s + 4-s − 5-s + 2.74·6-s + 3.45·7-s − 8-s + 4.51·9-s + 10-s + 11-s − 2.74·12-s + 2.23·13-s − 3.45·14-s + 2.74·15-s + 16-s − 3.54·17-s − 4.51·18-s + 2.53·19-s − 20-s − 9.47·21-s − 22-s − 5.12·23-s + 2.74·24-s + 25-s − 2.23·26-s − 4.15·27-s + 3.45·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.58·3-s + 0.5·4-s − 0.447·5-s + 1.11·6-s + 1.30·7-s − 0.353·8-s + 1.50·9-s + 0.316·10-s + 0.301·11-s − 0.791·12-s + 0.620·13-s − 0.923·14-s + 0.707·15-s + 0.250·16-s − 0.860·17-s − 1.06·18-s + 0.580·19-s − 0.223·20-s − 2.06·21-s − 0.213·22-s − 1.06·23-s + 0.559·24-s + 0.200·25-s − 0.439·26-s − 0.800·27-s + 0.652·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8909585952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8909585952\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.74T + 3T^{2} \) |
| 7 | \( 1 - 3.45T + 7T^{2} \) |
| 13 | \( 1 - 2.23T + 13T^{2} \) |
| 17 | \( 1 + 3.54T + 17T^{2} \) |
| 19 | \( 1 - 2.53T + 19T^{2} \) |
| 23 | \( 1 + 5.12T + 23T^{2} \) |
| 29 | \( 1 - 7.42T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 7.76T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 47 | \( 1 - 5.73T + 47T^{2} \) |
| 53 | \( 1 - 6.42T + 53T^{2} \) |
| 59 | \( 1 - 1.14T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 0.434T + 67T^{2} \) |
| 71 | \( 1 - 1.77T + 71T^{2} \) |
| 73 | \( 1 - 6.98T + 73T^{2} \) |
| 79 | \( 1 - 2.60T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 4.60T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.185090051065085552822696298245, −7.68141552876386156233182169884, −6.70644014548093320956660099694, −6.25205125870422703849568670461, −5.46982314985238398952129674102, −4.58626830405630675786368414500, −4.17245470622591669311522040874, −2.62946114131836001899833530550, −1.39845531970726444715151258968, −0.71154188317440190086564097289,
0.71154188317440190086564097289, 1.39845531970726444715151258968, 2.62946114131836001899833530550, 4.17245470622591669311522040874, 4.58626830405630675786368414500, 5.46982314985238398952129674102, 6.25205125870422703849568670461, 6.70644014548093320956660099694, 7.68141552876386156233182169884, 8.185090051065085552822696298245
