L(s) = 1 | + 2-s − 3-s + 4-s + 3.82·5-s − 6-s + 2.89·7-s + 8-s + 9-s + 3.82·10-s − 1.58·11-s − 12-s + 4.70·13-s + 2.89·14-s − 3.82·15-s + 16-s − 4.73·17-s + 18-s + 2.14·19-s + 3.82·20-s − 2.89·21-s − 1.58·22-s − 24-s + 9.61·25-s + 4.70·26-s − 27-s + 2.89·28-s − 0.588·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.70·5-s − 0.408·6-s + 1.09·7-s + 0.353·8-s + 0.333·9-s + 1.20·10-s − 0.476·11-s − 0.288·12-s + 1.30·13-s + 0.773·14-s − 0.987·15-s + 0.250·16-s − 1.14·17-s + 0.235·18-s + 0.492·19-s + 0.854·20-s − 0.631·21-s − 0.336·22-s − 0.204·24-s + 1.92·25-s + 0.923·26-s − 0.192·27-s + 0.546·28-s − 0.109·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.039035998\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.039035998\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - 3.82T + 5T^{2} \) |
| 7 | \( 1 - 2.89T + 7T^{2} \) |
| 11 | \( 1 + 1.58T + 11T^{2} \) |
| 13 | \( 1 - 4.70T + 13T^{2} \) |
| 17 | \( 1 + 4.73T + 17T^{2} \) |
| 19 | \( 1 - 2.14T + 19T^{2} \) |
| 29 | \( 1 + 0.588T + 29T^{2} \) |
| 31 | \( 1 - 0.170T + 31T^{2} \) |
| 37 | \( 1 - 4.81T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 5.45T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 1.90T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 5.87T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 0.113T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 0.873T + 97T^{2} \) |
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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
−8.750059625141416077314355439888, −7.83353644918867098351143738889, −6.87474490998796393406251352750, −6.08380442760485243754766408569, −5.72503650876245686116739784682, −4.92783476074500827060415491045, −4.30634250091168137008707925712, −2.95189047728993086517419247383, −1.95957502555241921865406795984, −1.29846118976158710537875400389,
1.29846118976158710537875400389, 1.95957502555241921865406795984, 2.95189047728993086517419247383, 4.30634250091168137008707925712, 4.92783476074500827060415491045, 5.72503650876245686116739784682, 6.08380442760485243754766408569, 6.87474490998796393406251352750, 7.83353644918867098351143738889, 8.750059625141416077314355439888