L(s) = 1 | + 0.137·3-s − 0.136·5-s + 0.729·7-s − 2.98·9-s − 11-s + 6.58·13-s − 0.0187·15-s + 7.50·17-s − 5.34·19-s + 0.100·21-s + 0.123·23-s − 4.98·25-s − 0.824·27-s − 5.30·29-s + 1.36·31-s − 0.137·33-s − 0.0992·35-s + 5.81·37-s + 0.907·39-s + 1.18·41-s + 8.07·43-s + 0.405·45-s + 12.6·47-s − 6.46·49-s + 1.03·51-s − 6.63·53-s + 0.136·55-s + ⋯ |
L(s) = 1 | + 0.0796·3-s − 0.0608·5-s + 0.275·7-s − 0.993·9-s − 0.301·11-s + 1.82·13-s − 0.00484·15-s + 1.81·17-s − 1.22·19-s + 0.0219·21-s + 0.0256·23-s − 0.996·25-s − 0.158·27-s − 0.984·29-s + 0.244·31-s − 0.0240·33-s − 0.0167·35-s + 0.956·37-s + 0.145·39-s + 0.184·41-s + 1.23·43-s + 0.0604·45-s + 1.84·47-s − 0.924·49-s + 0.144·51-s − 0.910·53-s + 0.0183·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.011297609\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.011297609\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 137 | \( 1 + T \) |
good | 3 | \( 1 - 0.137T + 3T^{2} \) |
| 5 | \( 1 + 0.136T + 5T^{2} \) |
| 7 | \( 1 - 0.729T + 7T^{2} \) |
| 13 | \( 1 - 6.58T + 13T^{2} \) |
| 17 | \( 1 - 7.50T + 17T^{2} \) |
| 19 | \( 1 + 5.34T + 19T^{2} \) |
| 23 | \( 1 - 0.123T + 23T^{2} \) |
| 29 | \( 1 + 5.30T + 29T^{2} \) |
| 31 | \( 1 - 1.36T + 31T^{2} \) |
| 37 | \( 1 - 5.81T + 37T^{2} \) |
| 41 | \( 1 - 1.18T + 41T^{2} \) |
| 43 | \( 1 - 8.07T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 6.63T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 3.68T + 61T^{2} \) |
| 67 | \( 1 + 5.69T + 67T^{2} \) |
| 71 | \( 1 - 1.50T + 71T^{2} \) |
| 73 | \( 1 + 4.39T + 73T^{2} \) |
| 79 | \( 1 - 9.92T + 79T^{2} \) |
| 83 | \( 1 + 7.47T + 83T^{2} \) |
| 89 | \( 1 - 4.25T + 89T^{2} \) |
| 97 | \( 1 - 4.65T + 97T^{2} \) |
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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
−7.87207357712498355739660164421, −7.79644001811232066924805953779, −6.46680285132372573422849493306, −5.80980244056246006198995855517, −5.53327561580204852505904711900, −4.26313262427671916784090113899, −3.66043486111130336696576904777, −2.85748962345573258074822603765, −1.83235125561897183692526121307, −0.75398947928626246562566744221,
0.75398947928626246562566744221, 1.83235125561897183692526121307, 2.85748962345573258074822603765, 3.66043486111130336696576904777, 4.26313262427671916784090113899, 5.53327561580204852505904711900, 5.80980244056246006198995855517, 6.46680285132372573422849493306, 7.79644001811232066924805953779, 7.87207357712498355739660164421