L(s) = 1 | − 3-s + 5-s + 1.56·7-s + 9-s + 2.43·11-s + 13-s − 15-s − 6.68·17-s − 3.12·19-s − 1.56·21-s − 4.68·23-s + 25-s − 27-s + 1.12·29-s + 8·31-s − 2.43·33-s + 1.56·35-s − 7.56·37-s − 39-s + 7.56·41-s + 0.876·43-s + 45-s + 10.2·47-s − 4.56·49-s + 6.68·51-s + 9.80·53-s + 2.43·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.590·7-s + 0.333·9-s + 0.735·11-s + 0.277·13-s − 0.258·15-s − 1.62·17-s − 0.716·19-s − 0.340·21-s − 0.976·23-s + 0.200·25-s − 0.192·27-s + 0.208·29-s + 1.43·31-s − 0.424·33-s + 0.263·35-s − 1.24·37-s − 0.160·39-s + 1.18·41-s + 0.133·43-s + 0.149·45-s + 1.49·47-s − 0.651·49-s + 0.936·51-s + 1.34·53-s + 0.328·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.851840644\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.851840644\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + 4.68T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 - 7.56T + 41T^{2} \) |
| 43 | \( 1 - 0.876T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 9.80T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 3.56T + 61T^{2} \) |
| 67 | \( 1 + 2.24T + 67T^{2} \) |
| 71 | \( 1 - 0.684T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 - 8.68T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 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
−8.224097650513623417916496989465, −7.09161401025374213447599277293, −6.60292610543465565502937808051, −5.98606951897659707621212482598, −5.23749658663981620327027064686, −4.32990203052017519126486896387, −3.98785295844212236400637331180, −2.51071611915224622095461593438, −1.84881406128631100305272600959, −0.74009342321396118555774292392,
0.74009342321396118555774292392, 1.84881406128631100305272600959, 2.51071611915224622095461593438, 3.98785295844212236400637331180, 4.32990203052017519126486896387, 5.23749658663981620327027064686, 5.98606951897659707621212482598, 6.60292610543465565502937808051, 7.09161401025374213447599277293, 8.224097650513623417916496989465