| L(s) = 1 | + 2-s + 3-s + 4-s + 2.95·5-s + 6-s + 2.84·7-s + 8-s + 9-s + 2.95·10-s + 3.27·11-s + 12-s − 13-s + 2.84·14-s + 2.95·15-s + 16-s − 4.61·17-s + 18-s + 2.77·19-s + 2.95·20-s + 2.84·21-s + 3.27·22-s − 8.73·23-s + 24-s + 3.71·25-s − 26-s + 27-s + 2.84·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.32·5-s + 0.408·6-s + 1.07·7-s + 0.353·8-s + 0.333·9-s + 0.933·10-s + 0.987·11-s + 0.288·12-s − 0.277·13-s + 0.760·14-s + 0.762·15-s + 0.250·16-s − 1.11·17-s + 0.235·18-s + 0.637·19-s + 0.660·20-s + 0.620·21-s + 0.698·22-s − 1.82·23-s + 0.204·24-s + 0.743·25-s − 0.196·26-s + 0.192·27-s + 0.537·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.759468106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.759468106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 - 2.95T + 5T^{2} \) |
| 7 | \( 1 - 2.84T + 7T^{2} \) |
| 11 | \( 1 - 3.27T + 11T^{2} \) |
| 17 | \( 1 + 4.61T + 17T^{2} \) |
| 19 | \( 1 - 2.77T + 19T^{2} \) |
| 23 | \( 1 + 8.73T + 23T^{2} \) |
| 29 | \( 1 - 3.69T + 29T^{2} \) |
| 31 | \( 1 - 4.04T + 31T^{2} \) |
| 37 | \( 1 - 6.80T + 37T^{2} \) |
| 41 | \( 1 + 3.86T + 41T^{2} \) |
| 43 | \( 1 + 7.96T + 43T^{2} \) |
| 47 | \( 1 + 1.74T + 47T^{2} \) |
| 59 | \( 1 + 4.96T + 59T^{2} \) |
| 61 | \( 1 - 0.744T + 61T^{2} \) |
| 67 | \( 1 + 5.19T + 67T^{2} \) |
| 71 | \( 1 - 1.86T + 71T^{2} \) |
| 73 | \( 1 - 0.0164T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 0.598T + 83T^{2} \) |
| 89 | \( 1 + 18.2T + 89T^{2} \) |
| 97 | \( 1 + 3.71T + 97T^{2} \) |
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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
−8.348297749805887314741186932284, −7.75859535280373740279285973570, −6.65724017054372782858569758375, −6.28754984842349530937167211380, −5.36688249150834177654889158145, −4.62223565626830851594439125136, −3.99129499485774360693119631695, −2.80824857063606977775116988289, −2.00260719226170955392222272194, −1.43841438002682307294951629108,
1.43841438002682307294951629108, 2.00260719226170955392222272194, 2.80824857063606977775116988289, 3.99129499485774360693119631695, 4.62223565626830851594439125136, 5.36688249150834177654889158145, 6.28754984842349530937167211380, 6.65724017054372782858569758375, 7.75859535280373740279285973570, 8.348297749805887314741186932284
