L(s) = 1 | − 2.67·2-s − 3-s + 5.13·4-s − 5-s + 2.67·6-s + 1.81·7-s − 8.38·8-s + 9-s + 2.67·10-s − 1.97·11-s − 5.13·12-s − 13-s − 4.84·14-s + 15-s + 12.1·16-s + 1.97·17-s − 2.67·18-s − 6.10·19-s − 5.13·20-s − 1.81·21-s + 5.28·22-s − 9.16·23-s + 8.38·24-s + 25-s + 2.67·26-s − 27-s + 9.31·28-s + ⋯ |
L(s) = 1 | − 1.88·2-s − 0.577·3-s + 2.56·4-s − 0.447·5-s + 1.09·6-s + 0.684·7-s − 2.96·8-s + 0.333·9-s + 0.844·10-s − 0.595·11-s − 1.48·12-s − 0.277·13-s − 1.29·14-s + 0.258·15-s + 3.03·16-s + 0.478·17-s − 0.629·18-s − 1.40·19-s − 1.14·20-s − 0.395·21-s + 1.12·22-s − 1.91·23-s + 1.71·24-s + 0.200·25-s + 0.523·26-s − 0.192·27-s + 1.75·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2880997951\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2880997951\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 7 | \( 1 - 1.81T + 7T^{2} \) |
| 11 | \( 1 + 1.97T + 11T^{2} \) |
| 17 | \( 1 - 1.97T + 17T^{2} \) |
| 19 | \( 1 + 6.10T + 19T^{2} \) |
| 23 | \( 1 + 9.16T + 23T^{2} \) |
| 29 | \( 1 - 6.88T + 29T^{2} \) |
| 37 | \( 1 + 0.595T + 37T^{2} \) |
| 41 | \( 1 + 9.47T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 + 5.27T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 2.25T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 6.57T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 6.21T + 83T^{2} \) |
| 89 | \( 1 - 2.65T + 89T^{2} \) |
| 97 | \( 1 - 5.91T + 97T^{2} \) |
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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
−8.153592878320785107405241035976, −7.78920154619274640326411432474, −6.65079270446224917178021421550, −6.48449296524177794662684553582, −5.37926109181727173319017567972, −4.54966471712268366517537219801, −3.38335244948215837234835996979, −2.24856508362856006968791459778, −1.61056418456049944682321235807, −0.38324712578409267005413829400,
0.38324712578409267005413829400, 1.61056418456049944682321235807, 2.24856508362856006968791459778, 3.38335244948215837234835996979, 4.54966471712268366517537219801, 5.37926109181727173319017567972, 6.48449296524177794662684553582, 6.65079270446224917178021421550, 7.78920154619274640326411432474, 8.153592878320785107405241035976