L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2.96·7-s + 8-s + 9-s − 3.35·11-s − 12-s + 4.96·13-s − 2.96·14-s + 16-s − 1.35·17-s + 18-s − 4.96·19-s + 2.96·21-s − 3.35·22-s + 23-s − 24-s + 4.96·26-s − 27-s − 2.96·28-s + 7.73·29-s − 4·31-s + 32-s + 3.35·33-s − 1.35·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.11·7-s + 0.353·8-s + 0.333·9-s − 1.01·11-s − 0.288·12-s + 1.37·13-s − 0.791·14-s + 0.250·16-s − 0.327·17-s + 0.235·18-s − 1.13·19-s + 0.646·21-s − 0.714·22-s + 0.208·23-s − 0.204·24-s + 0.973·26-s − 0.192·27-s − 0.559·28-s + 1.43·29-s − 0.718·31-s + 0.176·32-s + 0.583·33-s − 0.231·34-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
\(L(1)\) |
\(\approx\) |
\(1.909703014\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.909703014\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.96T + 7T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 - 4.96T + 13T^{2} \) |
| 17 | \( 1 + 1.35T + 17T^{2} \) |
| 19 | \( 1 + 4.96T + 19T^{2} \) |
| 29 | \( 1 - 7.73T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 - 4.70T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 3.22T + 47T^{2} \) |
| 53 | \( 1 - 6.96T + 53T^{2} \) |
| 59 | \( 1 - 1.22T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 7.61T + 67T^{2} \) |
| 71 | \( 1 + 2.18T + 71T^{2} \) |
| 73 | \( 1 - 9.92T + 73T^{2} \) |
| 79 | \( 1 + 4.12T + 79T^{2} \) |
| 83 | \( 1 - 6.38T + 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.545521495319039959968598527566, −7.71611285818214762915102170944, −6.79147877174990468956288952741, −6.18235319323705270727623727729, −5.78425592947165822186236168181, −4.72280742586250913612426821968, −4.01813165253729848860822729333, −3.12321174842808345853439281654, −2.25473465911943442873994133452, −0.74009342190394486475844744794,
0.74009342190394486475844744794, 2.25473465911943442873994133452, 3.12321174842808345853439281654, 4.01813165253729848860822729333, 4.72280742586250913612426821968, 5.78425592947165822186236168181, 6.18235319323705270727623727729, 6.79147877174990468956288952741, 7.71611285818214762915102170944, 8.545521495319039959968598527566