| L(s) = 1 | − 1.30·2-s − 3-s − 0.284·4-s + 5-s + 1.30·6-s + 3.58·7-s + 2.99·8-s + 9-s − 1.30·10-s − 2.84·11-s + 0.284·12-s + 5.52·13-s − 4.69·14-s − 15-s − 3.34·16-s − 1.00·17-s − 1.30·18-s + 4.45·19-s − 0.284·20-s − 3.58·21-s + 3.72·22-s − 2.99·24-s + 25-s − 7.24·26-s − 27-s − 1.02·28-s − 6.25·29-s + ⋯ |
| L(s) = 1 | − 0.926·2-s − 0.577·3-s − 0.142·4-s + 0.447·5-s + 0.534·6-s + 1.35·7-s + 1.05·8-s + 0.333·9-s − 0.414·10-s − 0.857·11-s + 0.0822·12-s + 1.53·13-s − 1.25·14-s − 0.258·15-s − 0.837·16-s − 0.243·17-s − 0.308·18-s + 1.02·19-s − 0.0636·20-s − 0.781·21-s + 0.793·22-s − 0.610·24-s + 0.200·25-s − 1.42·26-s − 0.192·27-s − 0.192·28-s − 1.16·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.310875140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.310875140\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 7 | \( 1 - 3.58T + 7T^{2} \) |
| 11 | \( 1 + 2.84T + 11T^{2} \) |
| 13 | \( 1 - 5.52T + 13T^{2} \) |
| 17 | \( 1 + 1.00T + 17T^{2} \) |
| 19 | \( 1 - 4.45T + 19T^{2} \) |
| 29 | \( 1 + 6.25T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 - 6.60T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 4.85T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 7.03T + 53T^{2} \) |
| 59 | \( 1 - 1.07T + 59T^{2} \) |
| 61 | \( 1 - 4.78T + 61T^{2} \) |
| 67 | \( 1 - 9.82T + 67T^{2} \) |
| 71 | \( 1 + 2.41T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 7.52T + 79T^{2} \) |
| 83 | \( 1 - 9.32T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−8.024563860604702674281544212317, −7.36860628768930382024191230719, −6.56335622332387816499911832856, −5.55074738682464322365521618461, −5.23606491942056448281491854758, −4.44359084177691657073940059302, −3.62961941051833966668787029004, −2.26690329022970987539370820779, −1.43864819944149587707347908737, −0.76456380241614479362240915615,
0.76456380241614479362240915615, 1.43864819944149587707347908737, 2.26690329022970987539370820779, 3.62961941051833966668787029004, 4.44359084177691657073940059302, 5.23606491942056448281491854758, 5.55074738682464322365521618461, 6.56335622332387816499911832856, 7.36860628768930382024191230719, 8.024563860604702674281544212317
