L(s) = 1 | − 2-s − 3-s + 4-s + 3.58·5-s + 6-s − 2.07·7-s − 8-s + 9-s − 3.58·10-s − 4.55·11-s − 12-s − 5.49·13-s + 2.07·14-s − 3.58·15-s + 16-s + 17-s − 18-s + 4.37·19-s + 3.58·20-s + 2.07·21-s + 4.55·22-s + 7.83·23-s + 24-s + 7.86·25-s + 5.49·26-s − 27-s − 2.07·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.60·5-s + 0.408·6-s − 0.782·7-s − 0.353·8-s + 0.333·9-s − 1.13·10-s − 1.37·11-s − 0.288·12-s − 1.52·13-s + 0.553·14-s − 0.926·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 1.00·19-s + 0.802·20-s + 0.451·21-s + 0.971·22-s + 1.63·23-s + 0.204·24-s + 1.57·25-s + 1.07·26-s − 0.192·27-s − 0.391·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.035674353\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.035674353\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 - 3.58T + 5T^{2} \) |
| 7 | \( 1 + 2.07T + 7T^{2} \) |
| 11 | \( 1 + 4.55T + 11T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 19 | \( 1 - 4.37T + 19T^{2} \) |
| 23 | \( 1 - 7.83T + 23T^{2} \) |
| 29 | \( 1 + 2.79T + 29T^{2} \) |
| 31 | \( 1 + 2.56T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 0.724T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 0.730T + 47T^{2} \) |
| 53 | \( 1 - 1.71T + 53T^{2} \) |
| 61 | \( 1 + 1.71T + 61T^{2} \) |
| 67 | \( 1 + 7.85T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 5.20T + 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 + 6.72T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.007490395224570252801016395310, −7.11639568239354579500636942077, −6.91700758323426861458482611492, −5.84144123711035923075304139974, −5.34122261627283271406154689110, −4.94211419556030341128778865563, −3.24333124789373198610313730947, −2.60324552372394777967143358872, −1.82046215184681348639973338087, −0.59307268374123515490891182617,
0.59307268374123515490891182617, 1.82046215184681348639973338087, 2.60324552372394777967143358872, 3.24333124789373198610313730947, 4.94211419556030341128778865563, 5.34122261627283271406154689110, 5.84144123711035923075304139974, 6.91700758323426861458482611492, 7.11639568239354579500636942077, 8.007490395224570252801016395310