| L(s) = 1 | + 2-s − 3-s + 4-s − 0.676·5-s − 6-s − 2.72·7-s + 8-s + 9-s − 0.676·10-s − 5.27·11-s − 12-s − 13-s − 2.72·14-s + 0.676·15-s + 16-s + 1.01·17-s + 18-s + 4.84·19-s − 0.676·20-s + 2.72·21-s − 5.27·22-s − 5.02·23-s − 24-s − 4.54·25-s − 26-s − 27-s − 2.72·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.302·5-s − 0.408·6-s − 1.02·7-s + 0.353·8-s + 0.333·9-s − 0.213·10-s − 1.58·11-s − 0.288·12-s − 0.277·13-s − 0.727·14-s + 0.174·15-s + 0.250·16-s + 0.246·17-s + 0.235·18-s + 1.11·19-s − 0.151·20-s + 0.593·21-s − 1.12·22-s − 1.04·23-s − 0.204·24-s − 0.908·25-s − 0.196·26-s − 0.192·27-s − 0.514·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.242745015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242745015\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 0.676T + 5T^{2} \) |
| 7 | \( 1 + 2.72T + 7T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 17 | \( 1 - 1.01T + 17T^{2} \) |
| 19 | \( 1 - 4.84T + 19T^{2} \) |
| 23 | \( 1 + 5.02T + 23T^{2} \) |
| 29 | \( 1 - 6.30T + 29T^{2} \) |
| 31 | \( 1 + 3.04T + 31T^{2} \) |
| 37 | \( 1 + 1.00T + 37T^{2} \) |
| 41 | \( 1 - 9.67T + 41T^{2} \) |
| 43 | \( 1 + 8.83T + 43T^{2} \) |
| 47 | \( 1 + 8.64T + 47T^{2} \) |
| 53 | \( 1 + 2.36T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 2.97T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 1.34T + 73T^{2} \) |
| 79 | \( 1 + 5.10T + 79T^{2} \) |
| 83 | \( 1 + 0.744T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 2.57T + 97T^{2} \) |
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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
−7.908601030893296992915499339553, −6.96108397264266839197697469883, −6.37010191384511330822561537795, −5.61603157078551500559581655614, −5.16867966948722730030942017657, −4.37145818529446937236489372767, −3.45535061631975882171926686890, −2.91735356066255310818460787211, −1.94955754800537447183933485177, −0.48221614469536022337110271216,
0.48221614469536022337110271216, 1.94955754800537447183933485177, 2.91735356066255310818460787211, 3.45535061631975882171926686890, 4.37145818529446937236489372767, 5.16867966948722730030942017657, 5.61603157078551500559581655614, 6.37010191384511330822561537795, 6.96108397264266839197697469883, 7.908601030893296992915499339553
