L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 12-s − 2·13-s − 14-s + 16-s − 3·17-s − 18-s + 2·19-s − 21-s + 23-s + 24-s + 2·26-s − 27-s + 28-s + 9·29-s − 10·31-s − 32-s + 3·34-s + 36-s − 11·37-s − 2·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.458·19-s − 0.218·21-s + 0.208·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.188·28-s + 1.67·29-s − 1.79·31-s − 0.176·32-s + 0.514·34-s + 1/6·36-s − 1.80·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9655854538\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9655854538\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.794600809253554108414317899065, −7.68951923417161112024477336194, −7.27458371577103265528132288120, −6.46371403904972115001924884841, −5.63756486048319970618648288723, −4.89264808151183218004848568224, −4.00414877783466423974431646684, −2.79082029760947265327739378277, −1.83281191114137763215272123388, −0.66650072722911527785273031528,
0.66650072722911527785273031528, 1.83281191114137763215272123388, 2.79082029760947265327739378277, 4.00414877783466423974431646684, 4.89264808151183218004848568224, 5.63756486048319970618648288723, 6.46371403904972115001924884841, 7.27458371577103265528132288120, 7.68951923417161112024477336194, 8.794600809253554108414317899065