| L(s) = 1 | − 1.81·2-s + 1.28·4-s − 2.52·7-s + 1.28·8-s − 0.813·11-s − 5.10·13-s + 4.57·14-s − 4.91·16-s + 3.39·17-s + 3.10·19-s + 1.47·22-s − 0.897·23-s + 9.25·26-s − 3.25·28-s − 4.44·29-s − 8.96·31-s + 6.33·32-s − 6.15·34-s − 2.08·37-s − 5.62·38-s − 41-s − 9.07·43-s − 1.04·44-s + 1.62·46-s − 0.235·47-s − 0.627·49-s − 6.57·52-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 0.644·4-s − 0.954·7-s + 0.455·8-s − 0.245·11-s − 1.41·13-s + 1.22·14-s − 1.22·16-s + 0.822·17-s + 0.711·19-s + 0.314·22-s − 0.187·23-s + 1.81·26-s − 0.615·28-s − 0.824·29-s − 1.61·31-s + 1.12·32-s − 1.05·34-s − 0.342·37-s − 0.912·38-s − 0.156·41-s − 1.38·43-s − 0.158·44-s + 0.239·46-s − 0.0343·47-s − 0.0896·49-s − 0.912·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3345095784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3345095784\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 11 | \( 1 + 0.813T + 11T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 - 3.39T + 17T^{2} \) |
| 19 | \( 1 - 3.10T + 19T^{2} \) |
| 23 | \( 1 + 0.897T + 23T^{2} \) |
| 29 | \( 1 + 4.44T + 29T^{2} \) |
| 31 | \( 1 + 8.96T + 31T^{2} \) |
| 37 | \( 1 + 2.08T + 37T^{2} \) |
| 43 | \( 1 + 9.07T + 43T^{2} \) |
| 47 | \( 1 + 0.235T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 1.04T + 59T^{2} \) |
| 61 | \( 1 - 1.91T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 4.75T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 7.68T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 2.72T + 97T^{2} \) |
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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
−7.77701355587114152902945603701, −7.09987467604896507135226875208, −6.84928184668657374640991123510, −5.56438355142999143150198882961, −5.21634770389184831922944835182, −4.09589560541481474038327990688, −3.29684683392443006859043035286, −2.42148391429629765640118417018, −1.53726485356250422841761134497, −0.34034279289204913458372212611,
0.34034279289204913458372212611, 1.53726485356250422841761134497, 2.42148391429629765640118417018, 3.29684683392443006859043035286, 4.09589560541481474038327990688, 5.21634770389184831922944835182, 5.56438355142999143150198882961, 6.84928184668657374640991123510, 7.09987467604896507135226875208, 7.77701355587114152902945603701
