| L(s) = 1 | − 2-s + 3-s + 4-s + 2.89·5-s − 6-s − 8-s + 9-s − 2.89·10-s + 1.61·11-s + 12-s + 3.02·13-s + 2.89·15-s + 16-s − 5.54·17-s − 18-s − 2.76·19-s + 2.89·20-s − 1.61·22-s + 23-s − 24-s + 3.39·25-s − 3.02·26-s + 27-s + 1.76·29-s − 2.89·30-s + 1.60·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.29·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.916·10-s + 0.488·11-s + 0.288·12-s + 0.839·13-s + 0.748·15-s + 0.250·16-s − 1.34·17-s − 0.235·18-s − 0.635·19-s + 0.647·20-s − 0.345·22-s + 0.208·23-s − 0.204·24-s + 0.678·25-s − 0.593·26-s + 0.192·27-s + 0.328·29-s − 0.528·30-s + 0.288·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.533071292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.533071292\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 2.89T + 5T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 - 3.02T + 13T^{2} \) |
| 17 | \( 1 + 5.54T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 - 1.60T + 31T^{2} \) |
| 37 | \( 1 + 1.02T + 37T^{2} \) |
| 41 | \( 1 + 6.76T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 4.15T + 47T^{2} \) |
| 53 | \( 1 - 4.89T + 53T^{2} \) |
| 59 | \( 1 - 6.95T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 - 4.87T + 71T^{2} \) |
| 73 | \( 1 + 5.13T + 73T^{2} \) |
| 79 | \( 1 + 6.89T + 79T^{2} \) |
| 83 | \( 1 + 5.15T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 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
−8.278609989684479561063082196501, −7.21882309825035097343969288728, −6.58025207218024730688027354174, −6.15800689247040937652458205274, −5.26884156034182528557323251494, −4.28896861279596745583501426431, −3.43667702882843657373374427419, −2.33315765694758981814055319851, −1.95637716569164910594179834153, −0.902833255693116305210684387818,
0.902833255693116305210684387818, 1.95637716569164910594179834153, 2.33315765694758981814055319851, 3.43667702882843657373374427419, 4.28896861279596745583501426431, 5.26884156034182528557323251494, 6.15800689247040937652458205274, 6.58025207218024730688027354174, 7.21882309825035097343969288728, 8.278609989684479561063082196501
