| L(s) = 1 | − 2-s + 3-s + 4-s − 3.88·5-s − 6-s + 7-s − 8-s + 9-s + 3.88·10-s − 2.01·11-s + 12-s + 4.46·13-s − 14-s − 3.88·15-s + 16-s + 1.73·17-s − 18-s + 6.29·19-s − 3.88·20-s + 21-s + 2.01·22-s + 3.07·23-s − 24-s + 10.0·25-s − 4.46·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.73·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.22·10-s − 0.606·11-s + 0.288·12-s + 1.23·13-s − 0.267·14-s − 1.00·15-s + 0.250·16-s + 0.420·17-s − 0.235·18-s + 1.44·19-s − 0.867·20-s + 0.218·21-s + 0.428·22-s + 0.640·23-s − 0.204·24-s + 2.01·25-s − 0.875·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.365414067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.365414067\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 5 | \( 1 + 3.88T + 5T^{2} \) |
| 11 | \( 1 + 2.01T + 11T^{2} \) |
| 13 | \( 1 - 4.46T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 - 3.07T + 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 - 3.34T + 31T^{2} \) |
| 37 | \( 1 + 3.58T + 37T^{2} \) |
| 41 | \( 1 + 8.04T + 41T^{2} \) |
| 43 | \( 1 - 5.48T + 43T^{2} \) |
| 47 | \( 1 + 0.224T + 47T^{2} \) |
| 53 | \( 1 - 5.63T + 53T^{2} \) |
| 59 | \( 1 - 6.71T + 59T^{2} \) |
| 61 | \( 1 - 0.0358T + 61T^{2} \) |
| 67 | \( 1 + 8.81T + 67T^{2} \) |
| 71 | \( 1 - 2.99T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 8.53T + 79T^{2} \) |
| 83 | \( 1 + 4.06T + 83T^{2} \) |
| 89 | \( 1 + 3.76T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−7.84318787470414881333869846425, −7.41513056849361739054468648277, −6.91321657517578683038731728283, −5.74973922912633129505060524922, −4.98465088876727999206101924196, −4.02644910061506429382765577454, −3.42471935020169764256420632931, −2.85808540498912930499317120361, −1.52384397602718258370308563906, −0.66606377134791720454122336989,
0.66606377134791720454122336989, 1.52384397602718258370308563906, 2.85808540498912930499317120361, 3.42471935020169764256420632931, 4.02644910061506429382765577454, 4.98465088876727999206101924196, 5.74973922912633129505060524922, 6.91321657517578683038731728283, 7.41513056849361739054468648277, 7.84318787470414881333869846425
