| L(s) = 1 | − 3-s − 2.12·7-s + 9-s + 3.15·11-s + 13-s + 0.515·17-s + 6.73·19-s + 2.12·21-s − 4.24·23-s − 27-s + 0.0302·29-s + 7.15·31-s − 3.15·33-s + 5.76·37-s − 39-s − 3.76·41-s − 3.28·43-s − 4.60·47-s − 2.48·49-s − 0.515·51-s + 0.280·53-s − 6.73·57-s + 11.3·59-s − 5.01·61-s − 2.12·63-s + 15.1·67-s + 4.24·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.803·7-s + 0.333·9-s + 0.951·11-s + 0.277·13-s + 0.124·17-s + 1.54·19-s + 0.463·21-s − 0.886·23-s − 0.192·27-s + 0.00562·29-s + 1.28·31-s − 0.549·33-s + 0.947·37-s − 0.160·39-s − 0.587·41-s − 0.500·43-s − 0.672·47-s − 0.354·49-s − 0.0721·51-s + 0.0384·53-s − 0.892·57-s + 1.47·59-s − 0.642·61-s − 0.267·63-s + 1.84·67-s + 0.511·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.585907154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.585907154\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 - 3.15T + 11T^{2} \) |
| 17 | \( 1 - 0.515T + 17T^{2} \) |
| 19 | \( 1 - 6.73T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 - 0.0302T + 29T^{2} \) |
| 31 | \( 1 - 7.15T + 31T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 + 3.76T + 41T^{2} \) |
| 43 | \( 1 + 3.28T + 43T^{2} \) |
| 47 | \( 1 + 4.60T + 47T^{2} \) |
| 53 | \( 1 - 0.280T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 5.01T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 2.23T + 71T^{2} \) |
| 73 | \( 1 + 2.96T + 73T^{2} \) |
| 79 | \( 1 + 8.98T + 79T^{2} \) |
| 83 | \( 1 - 5.40T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 0.909T + 97T^{2} \) |
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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
−7.84765365525870759820662349900, −6.91287890109926689397638981891, −6.52235762616392691878129591883, −5.84555850954056567130427870281, −5.15778848764940520258695333935, −4.25420808248985440632632188487, −3.57445013550573118374694001865, −2.81491138345144566489680797618, −1.56821581285309629810272338961, −0.68108905407635264500246216637,
0.68108905407635264500246216637, 1.56821581285309629810272338961, 2.81491138345144566489680797618, 3.57445013550573118374694001865, 4.25420808248985440632632188487, 5.15778848764940520258695333935, 5.84555850954056567130427870281, 6.52235762616392691878129591883, 6.91287890109926689397638981891, 7.84765365525870759820662349900
