L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2.61·7-s − 8-s + 9-s − 0.285·11-s − 12-s + 2.80·13-s + 2.61·14-s + 16-s + 6.49·17-s − 18-s − 0.443·19-s + 2.61·21-s + 0.285·22-s − 6.52·23-s + 24-s − 2.80·26-s − 27-s − 2.61·28-s + 7.25·29-s + 3.62·31-s − 32-s + 0.285·33-s − 6.49·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.990·7-s − 0.353·8-s + 0.333·9-s − 0.0861·11-s − 0.288·12-s + 0.778·13-s + 0.700·14-s + 0.250·16-s + 1.57·17-s − 0.235·18-s − 0.101·19-s + 0.571·21-s + 0.0609·22-s − 1.36·23-s + 0.204·24-s − 0.550·26-s − 0.192·27-s − 0.495·28-s + 1.34·29-s + 0.651·31-s − 0.176·32-s + 0.0497·33-s − 1.11·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8610683474\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8610683474\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2.61T + 7T^{2} \) |
| 11 | \( 1 + 0.285T + 11T^{2} \) |
| 13 | \( 1 - 2.80T + 13T^{2} \) |
| 17 | \( 1 - 6.49T + 17T^{2} \) |
| 19 | \( 1 + 0.443T + 19T^{2} \) |
| 23 | \( 1 + 6.52T + 23T^{2} \) |
| 29 | \( 1 - 7.25T + 29T^{2} \) |
| 31 | \( 1 - 3.62T + 31T^{2} \) |
| 37 | \( 1 + 4.53T + 37T^{2} \) |
| 41 | \( 1 - 5.32T + 41T^{2} \) |
| 43 | \( 1 + 8.05T + 43T^{2} \) |
| 47 | \( 1 + 8.99T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 + 3.74T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 4.81T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 7.81T + 79T^{2} \) |
| 83 | \( 1 - 5.16T + 83T^{2} \) |
| 89 | \( 1 + 6.16T + 89T^{2} \) |
| 97 | \( 1 + 5.45T + 97T^{2} \) |
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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
−8.287145619723993970944644266086, −8.032179219835612953971868113865, −6.90606393553171994300824510176, −6.35411061789409492338513688273, −5.80976197774413000659883991368, −4.84395274002897862172318090023, −3.66458117111768151499408504019, −3.03997507207469277726267083097, −1.70751230159946075707467253181, −0.62707482916999882208419498011,
0.62707482916999882208419498011, 1.70751230159946075707467253181, 3.03997507207469277726267083097, 3.66458117111768151499408504019, 4.84395274002897862172318090023, 5.80976197774413000659883991368, 6.35411061789409492338513688273, 6.90606393553171994300824510176, 8.032179219835612953971868113865, 8.287145619723993970944644266086