| L(s) = 1 | − 3-s − 2·7-s + 9-s + 13-s − 6·19-s + 2·21-s + 2·23-s − 27-s − 4·29-s + 8·31-s + 10·37-s − 39-s − 2·41-s + 12·43-s + 4·47-s − 3·49-s − 6·53-s + 6·57-s + 8·59-s − 2·61-s − 2·63-s + 4·67-s − 2·69-s + 81-s − 4·83-s + 4·87-s − 14·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 1.37·19-s + 0.436·21-s + 0.417·23-s − 0.192·27-s − 0.742·29-s + 1.43·31-s + 1.64·37-s − 0.160·39-s − 0.312·41-s + 1.82·43-s + 0.583·47-s − 3/7·49-s − 0.824·53-s + 0.794·57-s + 1.04·59-s − 0.256·61-s − 0.251·63-s + 0.488·67-s − 0.240·69-s + 1/9·81-s − 0.439·83-s + 0.428·87-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.265225348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.265225348\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.30847113725513, −14.53984782035584, −14.08112265963005, −13.27186707287051, −12.92677627667016, −12.59121835366020, −11.88846576155999, −11.29450105924819, −10.86947038411979, −10.31962065549834, −9.711772702342714, −9.267866671561050, −8.585386473901437, −7.988750352530814, −7.342117053731704, −6.637844165166326, −6.236775622377739, −5.781776289074976, −4.978350030894133, −4.265863315560066, −3.870421035614966, −2.877048082972042, −2.356804618795567, −1.296982544862939, −0.4728239736373465,
0.4728239736373465, 1.296982544862939, 2.356804618795567, 2.877048082972042, 3.870421035614966, 4.265863315560066, 4.978350030894133, 5.781776289074976, 6.236775622377739, 6.637844165166326, 7.342117053731704, 7.988750352530814, 8.585386473901437, 9.267866671561050, 9.711772702342714, 10.31962065549834, 10.86947038411979, 11.29450105924819, 11.88846576155999, 12.59121835366020, 12.92677627667016, 13.27186707287051, 14.08112265963005, 14.53984782035584, 15.30847113725513
