| L(s) = 1 | − 2-s + 3-s + 4-s − 2.90·5-s − 6-s + 7-s − 8-s + 9-s + 2.90·10-s + 4.70·11-s + 12-s − 0.944·13-s − 14-s − 2.90·15-s + 16-s + 7.13·17-s − 18-s − 2.37·19-s − 2.90·20-s + 21-s − 4.70·22-s + 1.52·23-s − 24-s + 3.41·25-s + 0.944·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.29·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.917·10-s + 1.41·11-s + 0.288·12-s − 0.261·13-s − 0.267·14-s − 0.749·15-s + 0.250·16-s + 1.72·17-s − 0.235·18-s − 0.545·19-s − 0.648·20-s + 0.218·21-s − 1.00·22-s + 0.317·23-s − 0.204·24-s + 0.683·25-s + 0.185·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.761197155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.761197155\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 5 | \( 1 + 2.90T + 5T^{2} \) |
| 11 | \( 1 - 4.70T + 11T^{2} \) |
| 13 | \( 1 + 0.944T + 13T^{2} \) |
| 17 | \( 1 - 7.13T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 - 3.78T + 29T^{2} \) |
| 31 | \( 1 - 5.22T + 31T^{2} \) |
| 37 | \( 1 - 6.79T + 37T^{2} \) |
| 41 | \( 1 + 8.73T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 + 3.08T + 47T^{2} \) |
| 53 | \( 1 - 1.58T + 53T^{2} \) |
| 59 | \( 1 - 1.73T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 + 2.70T + 89T^{2} \) |
| 97 | \( 1 + 2.93T + 97T^{2} \) |
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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
−8.014124919251874305790653549908, −7.30311964485381813709261837620, −6.78560426097361236497142979564, −5.94188739276851811737783043160, −4.84484189334197891881371320250, −4.09869814210044053063093852939, −3.49309055893268142799202830464, −2.73232014853786186389908820005, −1.52055126463833092888312400630, −0.77336284352126104987767083472,
0.77336284352126104987767083472, 1.52055126463833092888312400630, 2.73232014853786186389908820005, 3.49309055893268142799202830464, 4.09869814210044053063093852939, 4.84484189334197891881371320250, 5.94188739276851811737783043160, 6.78560426097361236497142979564, 7.30311964485381813709261837620, 8.014124919251874305790653549908
