| L(s) = 1 | + 0.585·5-s + 3.41·7-s + 2.82·11-s + 7.41·17-s + 6.24·19-s − 23-s − 4.65·25-s + 8.48·29-s − 8.48·31-s + 2·35-s − 4.82·37-s − 1.65·41-s + 1.75·43-s + 0.343·47-s + 4.65·49-s − 5.07·53-s + 1.65·55-s − 7.65·59-s − 0.828·61-s − 8.58·67-s + 13.6·71-s + 13.3·73-s + 9.65·77-s − 7.89·79-s − 6.82·83-s + 4.34·85-s + 13.0·89-s + ⋯ |
| L(s) = 1 | + 0.261·5-s + 1.29·7-s + 0.852·11-s + 1.79·17-s + 1.43·19-s − 0.208·23-s − 0.931·25-s + 1.57·29-s − 1.52·31-s + 0.338·35-s − 0.793·37-s − 0.258·41-s + 0.267·43-s + 0.0500·47-s + 0.665·49-s − 0.696·53-s + 0.223·55-s − 0.996·59-s − 0.106·61-s − 1.04·67-s + 1.62·71-s + 1.55·73-s + 1.10·77-s − 0.888·79-s − 0.749·83-s + 0.471·85-s + 1.38·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.729520984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.729520984\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 7.41T + 17T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 + 4.82T + 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 - 0.343T + 47T^{2} \) |
| 53 | \( 1 + 5.07T + 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 + 0.828T + 61T^{2} \) |
| 67 | \( 1 + 8.58T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 7.89T + 79T^{2} \) |
| 83 | \( 1 + 6.82T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.557220034893780636279034245398, −7.80848632756913015296539196059, −7.37713881639831965555240641962, −6.30911655229730700302740946726, −5.46131557362256224494128063604, −4.98149109813446026477184831699, −3.90668258917506927623308138999, −3.13210903220216940211834365248, −1.78333588198611120633524440503, −1.12257476584875272809618309642,
1.12257476584875272809618309642, 1.78333588198611120633524440503, 3.13210903220216940211834365248, 3.90668258917506927623308138999, 4.98149109813446026477184831699, 5.46131557362256224494128063604, 6.30911655229730700302740946726, 7.37713881639831965555240641962, 7.80848632756913015296539196059, 8.557220034893780636279034245398
