L(s) = 1 | + 2.79·2-s + 2.29·3-s + 5.78·4-s + 6.39·6-s + 7-s + 10.5·8-s + 2.25·9-s + 0.340·11-s + 13.2·12-s + 1.30·13-s + 2.79·14-s + 17.9·16-s − 2.59·17-s + 6.29·18-s − 8.28·19-s + 2.29·21-s + 0.950·22-s − 23-s + 24.2·24-s + 3.63·26-s − 1.70·27-s + 5.78·28-s + 3.78·29-s − 9.08·31-s + 28.8·32-s + 0.780·33-s − 7.24·34-s + ⋯ |
L(s) = 1 | + 1.97·2-s + 1.32·3-s + 2.89·4-s + 2.61·6-s + 0.377·7-s + 3.73·8-s + 0.752·9-s + 0.102·11-s + 3.83·12-s + 0.361·13-s + 0.745·14-s + 4.48·16-s − 0.629·17-s + 1.48·18-s − 1.90·19-s + 0.500·21-s + 0.202·22-s − 0.208·23-s + 4.94·24-s + 0.713·26-s − 0.327·27-s + 1.09·28-s + 0.702·29-s − 1.63·31-s + 5.10·32-s + 0.135·33-s − 1.24·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(11.44680465\) |
\(L(\frac12)\) |
\(\approx\) |
\(11.44680465\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 2.79T + 2T^{2} \) |
| 3 | \( 1 - 2.29T + 3T^{2} \) |
| 11 | \( 1 - 0.340T + 11T^{2} \) |
| 13 | \( 1 - 1.30T + 13T^{2} \) |
| 17 | \( 1 + 2.59T + 17T^{2} \) |
| 19 | \( 1 + 8.28T + 19T^{2} \) |
| 29 | \( 1 - 3.78T + 29T^{2} \) |
| 31 | \( 1 + 9.08T + 31T^{2} \) |
| 37 | \( 1 + 6.40T + 37T^{2} \) |
| 41 | \( 1 - 0.841T + 41T^{2} \) |
| 43 | \( 1 + 9.17T + 43T^{2} \) |
| 47 | \( 1 + 4.66T + 47T^{2} \) |
| 53 | \( 1 + 3.49T + 53T^{2} \) |
| 59 | \( 1 + 2.98T + 59T^{2} \) |
| 61 | \( 1 + 3.00T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 5.87T + 71T^{2} \) |
| 73 | \( 1 - 7.81T + 73T^{2} \) |
| 79 | \( 1 - 4.35T + 79T^{2} \) |
| 83 | \( 1 - 9.31T + 83T^{2} \) |
| 89 | \( 1 - 2.83T + 89T^{2} \) |
| 97 | \( 1 + 0.660T + 97T^{2} \) |
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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
−8.281990274446470393074100810574, −7.59274810899672270103394885583, −6.71147695479232130148229212088, −6.25371431395458221769192518631, −5.20011842351236834169812497179, −4.52527435040696029220587942557, −3.74163579815364843621048384486, −3.28511450673417390023347974165, −2.12887047953908506280215159687, −1.90971705658335680258249652614,
1.90971705658335680258249652614, 2.12887047953908506280215159687, 3.28511450673417390023347974165, 3.74163579815364843621048384486, 4.52527435040696029220587942557, 5.20011842351236834169812497179, 6.25371431395458221769192518631, 6.71147695479232130148229212088, 7.59274810899672270103394885583, 8.281990274446470393074100810574