| L(s) = 1 | + 2-s − 3-s + 4-s − 0.869·5-s − 6-s + 3.97·7-s + 8-s + 9-s − 0.869·10-s + 3.41·11-s − 12-s + 3.88·13-s + 3.97·14-s + 0.869·15-s + 16-s − 1.10·17-s + 18-s + 3.97·19-s − 0.869·20-s − 3.97·21-s + 3.41·22-s + 23-s − 24-s − 4.24·25-s + 3.88·26-s − 27-s + 3.97·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.388·5-s − 0.408·6-s + 1.50·7-s + 0.353·8-s + 0.333·9-s − 0.275·10-s + 1.02·11-s − 0.288·12-s + 1.07·13-s + 1.06·14-s + 0.224·15-s + 0.250·16-s − 0.267·17-s + 0.235·18-s + 0.911·19-s − 0.194·20-s − 0.866·21-s + 0.727·22-s + 0.208·23-s − 0.204·24-s − 0.848·25-s + 0.762·26-s − 0.192·27-s + 0.750·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.349334136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.349334136\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 0.869T + 5T^{2} \) |
| 7 | \( 1 - 3.97T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 - 3.88T + 13T^{2} \) |
| 17 | \( 1 + 1.10T + 17T^{2} \) |
| 19 | \( 1 - 3.97T + 19T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 - 6.30T + 47T^{2} \) |
| 53 | \( 1 - 3.72T + 53T^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 - 8.38T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 3.08T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 0.168T + 79T^{2} \) |
| 83 | \( 1 + 7.07T + 83T^{2} \) |
| 89 | \( 1 - 6.49T + 89T^{2} \) |
| 97 | \( 1 + 0.491T + 97T^{2} \) |
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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
−8.406758408923459893120128221948, −7.57973190210378481679420900321, −6.90688716711925929392255360561, −6.09537798794970417396990177968, −5.42324909570342058504740865697, −4.60262574261817478154861361486, −4.10215985565995752891967226197, −3.19972358635168019337874589509, −1.78599504935920333233772536024, −1.09839432668947993786859312568,
1.09839432668947993786859312568, 1.78599504935920333233772536024, 3.19972358635168019337874589509, 4.10215985565995752891967226197, 4.60262574261817478154861361486, 5.42324909570342058504740865697, 6.09537798794970417396990177968, 6.90688716711925929392255360561, 7.57973190210378481679420900321, 8.406758408923459893120128221948
