| L(s) = 1 | + 3-s + 3.16·5-s − 3.98·7-s + 9-s − 6.19·11-s + 3.16·15-s + 0.0827·17-s + 2.80·19-s − 3.98·21-s + 6.47·23-s + 5.01·25-s + 27-s + 7.19·29-s + 4.21·31-s − 6.19·33-s − 12.6·35-s + 0.190·37-s − 2.55·41-s − 6.72·43-s + 3.16·45-s + 13.0·47-s + 8.86·49-s + 0.0827·51-s + 1.86·53-s − 19.5·55-s + 2.80·57-s − 4.92·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.41·5-s − 1.50·7-s + 0.333·9-s − 1.86·11-s + 0.816·15-s + 0.0200·17-s + 0.644·19-s − 0.869·21-s + 1.35·23-s + 1.00·25-s + 0.192·27-s + 1.33·29-s + 0.756·31-s − 1.07·33-s − 2.13·35-s + 0.0312·37-s − 0.399·41-s − 1.02·43-s + 0.471·45-s + 1.89·47-s + 1.26·49-s + 0.0115·51-s + 0.255·53-s − 2.64·55-s + 0.371·57-s − 0.641·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.421009029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.421009029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.16T + 5T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 11 | \( 1 + 6.19T + 11T^{2} \) |
| 17 | \( 1 - 0.0827T + 17T^{2} \) |
| 19 | \( 1 - 2.80T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 - 4.21T + 31T^{2} \) |
| 37 | \( 1 - 0.190T + 37T^{2} \) |
| 41 | \( 1 + 2.55T + 41T^{2} \) |
| 43 | \( 1 + 6.72T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 - 1.86T + 53T^{2} \) |
| 59 | \( 1 + 4.92T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 - 6.22T + 67T^{2} \) |
| 71 | \( 1 + 6.19T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 6.44T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−8.576142349505211775733803343421, −7.70516711843952141152504818171, −6.85082107553449063147145658403, −6.32102772484311841797992982879, −5.41035745159016185796820132852, −4.94450716821415809806367543006, −3.50658588176978326064733777632, −2.74401980514231676635807727137, −2.37081275213468629777437291038, −0.848993603573427775272206251365,
0.848993603573427775272206251365, 2.37081275213468629777437291038, 2.74401980514231676635807727137, 3.50658588176978326064733777632, 4.94450716821415809806367543006, 5.41035745159016185796820132852, 6.32102772484311841797992982879, 6.85082107553449063147145658403, 7.70516711843952141152504818171, 8.576142349505211775733803343421
