| L(s) = 1 | − 2.10·3-s + 5-s + 4.72·7-s + 1.41·9-s + 4.62·11-s − 4.47·13-s − 2.10·15-s − 5.66·17-s − 1.34·19-s − 9.93·21-s − 4.44·23-s + 25-s + 3.32·27-s − 4.27·29-s − 9.40·31-s − 9.72·33-s + 4.72·35-s + 6.89·37-s + 9.41·39-s + 5.69·41-s + 5.04·43-s + 1.41·45-s + 8.76·47-s + 15.3·49-s + 11.9·51-s − 8.23·53-s + 4.62·55-s + ⋯ |
| L(s) = 1 | − 1.21·3-s + 0.447·5-s + 1.78·7-s + 0.473·9-s + 1.39·11-s − 1.24·13-s − 0.542·15-s − 1.37·17-s − 0.308·19-s − 2.16·21-s − 0.926·23-s + 0.200·25-s + 0.639·27-s − 0.793·29-s − 1.68·31-s − 1.69·33-s + 0.799·35-s + 1.13·37-s + 1.50·39-s + 0.889·41-s + 0.769·43-s + 0.211·45-s + 1.27·47-s + 2.19·49-s + 1.66·51-s − 1.13·53-s + 0.624·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.539816792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.539816792\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 3 | \( 1 + 2.10T + 3T^{2} \) |
| 7 | \( 1 - 4.72T + 7T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 5.66T + 17T^{2} \) |
| 19 | \( 1 + 1.34T + 19T^{2} \) |
| 23 | \( 1 + 4.44T + 23T^{2} \) |
| 29 | \( 1 + 4.27T + 29T^{2} \) |
| 31 | \( 1 + 9.40T + 31T^{2} \) |
| 37 | \( 1 - 6.89T + 37T^{2} \) |
| 41 | \( 1 - 5.69T + 41T^{2} \) |
| 43 | \( 1 - 5.04T + 43T^{2} \) |
| 47 | \( 1 - 8.76T + 47T^{2} \) |
| 53 | \( 1 + 8.23T + 53T^{2} \) |
| 59 | \( 1 - 7.52T + 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 - 6.38T + 67T^{2} \) |
| 71 | \( 1 + 5.84T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 7.89T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 97 | \( 1 + 0.674T + 97T^{2} \) |
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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
−7.69018653356608346664067343026, −7.26551884709434922754208632207, −6.33660690977892316306694185999, −5.85251829453722416092082662407, −5.09064535539160617391682583259, −4.52582351234521965641324238514, −3.97965907219914818954758926310, −2.27962364673451117101689468761, −1.81804555777902448833193584236, −0.67708046553560873621793601163,
0.67708046553560873621793601163, 1.81804555777902448833193584236, 2.27962364673451117101689468761, 3.97965907219914818954758926310, 4.52582351234521965641324238514, 5.09064535539160617391682583259, 5.85251829453722416092082662407, 6.33660690977892316306694185999, 7.26551884709434922754208632207, 7.69018653356608346664067343026
