| L(s) = 1 | + 1.61·2-s + 3-s + 0.618·4-s − 4.23·5-s + 1.61·6-s − 7-s − 2.23·8-s + 9-s − 6.85·10-s + 0.618·12-s + 6.23·13-s − 1.61·14-s − 4.23·15-s − 4.85·16-s − 4.47·17-s + 1.61·18-s + 3·19-s − 2.61·20-s − 21-s + 8.47·23-s − 2.23·24-s + 12.9·25-s + 10.0·26-s + 27-s − 0.618·28-s − 3·29-s − 6.85·30-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.577·3-s + 0.309·4-s − 1.89·5-s + 0.660·6-s − 0.377·7-s − 0.790·8-s + 0.333·9-s − 2.16·10-s + 0.178·12-s + 1.72·13-s − 0.432·14-s − 1.09·15-s − 1.21·16-s − 1.08·17-s + 0.381·18-s + 0.688·19-s − 0.585·20-s − 0.218·21-s + 1.76·23-s − 0.456·24-s + 2.58·25-s + 1.97·26-s + 0.192·27-s − 0.116·28-s − 0.557·29-s − 1.25·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.433769256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.433769256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 + 4.23T + 5T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 3.47T + 37T^{2} \) |
| 41 | \( 1 - 1.52T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 8.94T + 53T^{2} \) |
| 59 | \( 1 + 1.47T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 + 8.70T + 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.857143704386704643850720887222, −8.170794739156512386351255390664, −7.24857412388667026438192650925, −6.66498985060980633707361761410, −5.64248568446366900615917976346, −4.58447984416195775358413899347, −4.02555076456381273504056017435, −3.43870567717536772254314128943, −2.79248104188882076909598089751, −0.814502883612581387160443608813,
0.814502883612581387160443608813, 2.79248104188882076909598089751, 3.43870567717536772254314128943, 4.02555076456381273504056017435, 4.58447984416195775358413899347, 5.64248568446366900615917976346, 6.66498985060980633707361761410, 7.24857412388667026438192650925, 8.170794739156512386351255390664, 8.857143704386704643850720887222
