| L(s) = 1 | − 2.34·3-s − 1.48·5-s − 7-s + 2.48·9-s + 11-s + 3.14·13-s + 3.48·15-s + 5.53·17-s − 6.97·19-s + 2.34·21-s + 3.48·23-s − 2.78·25-s + 1.19·27-s − 2.68·29-s + 4.63·31-s − 2.34·33-s + 1.48·35-s + 0.510·37-s − 7.37·39-s + 5.43·41-s − 11.3·43-s − 3.70·45-s − 8.81·47-s + 49-s − 12.9·51-s − 0.685·53-s − 1.48·55-s + ⋯ |
| L(s) = 1 | − 1.35·3-s − 0.666·5-s − 0.377·7-s + 0.829·9-s + 0.301·11-s + 0.872·13-s + 0.900·15-s + 1.34·17-s − 1.60·19-s + 0.511·21-s + 0.727·23-s − 0.556·25-s + 0.230·27-s − 0.498·29-s + 0.832·31-s − 0.407·33-s + 0.251·35-s + 0.0839·37-s − 1.18·39-s + 0.849·41-s − 1.73·43-s − 0.552·45-s − 1.28·47-s + 0.142·49-s − 1.81·51-s − 0.0942·53-s − 0.200·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7279908909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7279908909\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.34T + 3T^{2} \) |
| 5 | \( 1 + 1.48T + 5T^{2} \) |
| 13 | \( 1 - 3.14T + 13T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 + 2.68T + 29T^{2} \) |
| 31 | \( 1 - 4.63T + 31T^{2} \) |
| 37 | \( 1 - 0.510T + 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 8.81T + 47T^{2} \) |
| 53 | \( 1 + 0.685T + 53T^{2} \) |
| 59 | \( 1 - 2.92T + 59T^{2} \) |
| 61 | \( 1 + 3.83T + 61T^{2} \) |
| 67 | \( 1 + 9.88T + 67T^{2} \) |
| 71 | \( 1 - 4.51T + 71T^{2} \) |
| 73 | \( 1 + 0.518T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 6.39T + 83T^{2} \) |
| 89 | \( 1 - 4.21T + 89T^{2} \) |
| 97 | \( 1 - 1.48T + 97T^{2} \) |
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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
−8.211092563580651655993220736441, −7.45995101503470156621529722112, −6.49936563852000211667899968417, −6.23208134586424517864760314755, −5.41879485226456040757612067951, −4.62284778795509380947977511878, −3.85866313175302265132462307389, −3.07745812826845890424959988324, −1.58049609864733983718843623101, −0.51964376121547142971706527419,
0.51964376121547142971706527419, 1.58049609864733983718843623101, 3.07745812826845890424959988324, 3.85866313175302265132462307389, 4.62284778795509380947977511878, 5.41879485226456040757612067951, 6.23208134586424517864760314755, 6.49936563852000211667899968417, 7.45995101503470156621529722112, 8.211092563580651655993220736441
