| L(s) = 1 | + 2.67·2-s + 2.41·3-s + 5.16·4-s + 6.46·6-s + 0.976·7-s + 8.46·8-s + 2.83·9-s + 2.37·11-s + 12.4·12-s − 5.63·13-s + 2.61·14-s + 12.3·16-s + 4.16·17-s + 7.58·18-s − 1.76·19-s + 2.35·21-s + 6.36·22-s − 3.61·23-s + 20.4·24-s − 15.0·26-s − 0.399·27-s + 5.04·28-s + 4.75·29-s − 4.18·31-s + 16.0·32-s + 5.74·33-s + 11.1·34-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 1.39·3-s + 2.58·4-s + 2.63·6-s + 0.369·7-s + 2.99·8-s + 0.944·9-s + 0.716·11-s + 3.59·12-s − 1.56·13-s + 0.698·14-s + 3.08·16-s + 1.00·17-s + 1.78·18-s − 0.405·19-s + 0.514·21-s + 1.35·22-s − 0.753·23-s + 4.17·24-s − 2.96·26-s − 0.0769·27-s + 0.952·28-s + 0.883·29-s − 0.750·31-s + 2.84·32-s + 0.999·33-s + 1.91·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.06306466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.06306466\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 3 | \( 1 - 2.41T + 3T^{2} \) |
| 7 | \( 1 - 0.976T + 7T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 13 | \( 1 + 5.63T + 13T^{2} \) |
| 17 | \( 1 - 4.16T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 + 3.61T + 23T^{2} \) |
| 29 | \( 1 - 4.75T + 29T^{2} \) |
| 31 | \( 1 + 4.18T + 31T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 - 3.06T + 41T^{2} \) |
| 43 | \( 1 - 2.48T + 43T^{2} \) |
| 47 | \( 1 - 0.632T + 47T^{2} \) |
| 53 | \( 1 - 4.42T + 53T^{2} \) |
| 59 | \( 1 + 9.05T + 59T^{2} \) |
| 61 | \( 1 - 8.64T + 61T^{2} \) |
| 67 | \( 1 + 4.20T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 2.23T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 4.45T + 83T^{2} \) |
| 89 | \( 1 + 0.784T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69126673265501369509024768712, −7.44066809701685601856309134696, −6.55892555334700056447582209990, −5.77907145467418229458282487092, −4.98239614001110403307999069481, −4.30198686925377736377094462101, −3.70142051880815968503611978332, −2.91648344053166620817056207125, −2.34994986847421671573691023978, −1.56210859929171125531347283030,
1.56210859929171125531347283030, 2.34994986847421671573691023978, 2.91648344053166620817056207125, 3.70142051880815968503611978332, 4.30198686925377736377094462101, 4.98239614001110403307999069481, 5.77907145467418229458282487092, 6.55892555334700056447582209990, 7.44066809701685601856309134696, 7.69126673265501369509024768712
