L(s) = 1 | + 1.79·2-s − 3-s + 1.22·4-s + 0.0102·5-s − 1.79·6-s − 0.754·7-s − 1.39·8-s + 9-s + 0.0183·10-s + 5.42·11-s − 1.22·12-s − 1.45·13-s − 1.35·14-s − 0.0102·15-s − 4.95·16-s − 17-s + 1.79·18-s + 3.28·19-s + 0.0124·20-s + 0.754·21-s + 9.74·22-s − 4.49·23-s + 1.39·24-s − 4.99·25-s − 2.61·26-s − 27-s − 0.922·28-s + ⋯ |
L(s) = 1 | + 1.26·2-s − 0.577·3-s + 0.610·4-s + 0.00456·5-s − 0.732·6-s − 0.285·7-s − 0.493·8-s + 0.333·9-s + 0.00579·10-s + 1.63·11-s − 0.352·12-s − 0.404·13-s − 0.362·14-s − 0.00263·15-s − 1.23·16-s − 0.242·17-s + 0.423·18-s + 0.752·19-s + 0.00278·20-s + 0.164·21-s + 2.07·22-s − 0.936·23-s + 0.285·24-s − 0.999·25-s − 0.513·26-s − 0.192·27-s − 0.174·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 - 1.79T + 2T^{2} \) |
| 5 | \( 1 - 0.0102T + 5T^{2} \) |
| 7 | \( 1 + 0.754T + 7T^{2} \) |
| 11 | \( 1 - 5.42T + 11T^{2} \) |
| 13 | \( 1 + 1.45T + 13T^{2} \) |
| 19 | \( 1 - 3.28T + 19T^{2} \) |
| 23 | \( 1 + 4.49T + 23T^{2} \) |
| 29 | \( 1 + 7.70T + 29T^{2} \) |
| 31 | \( 1 - 1.34T + 31T^{2} \) |
| 37 | \( 1 - 6.85T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 4.83T + 43T^{2} \) |
| 47 | \( 1 + 8.18T + 47T^{2} \) |
| 53 | \( 1 - 7.35T + 53T^{2} \) |
| 59 | \( 1 + 1.35T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 4.70T + 71T^{2} \) |
| 73 | \( 1 + 7.69T + 73T^{2} \) |
| 79 | \( 1 - 6.14T + 79T^{2} \) |
| 83 | \( 1 + 9.19T + 83T^{2} \) |
| 89 | \( 1 - 5.29T + 89T^{2} \) |
| 97 | \( 1 + 6.42T + 97T^{2} \) |
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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
−7.25073846939583860282439388631, −6.44108696455711501189234530343, −5.98250936525426704135116297638, −5.50829677374680475425982923982, −4.45408588053896372913227964739, −4.12255984462079514152913177476, −3.43284503895104029624517401741, −2.43708888438311868172384514618, −1.38084809174983805918479564197, 0,
1.38084809174983805918479564197, 2.43708888438311868172384514618, 3.43284503895104029624517401741, 4.12255984462079514152913177476, 4.45408588053896372913227964739, 5.50829677374680475425982923982, 5.98250936525426704135116297638, 6.44108696455711501189234530343, 7.25073846939583860282439388631