| L(s) = 1 | + 2-s + 3-s + 4-s + 0.380·5-s + 6-s − 2.38·7-s + 8-s + 9-s + 0.380·10-s + 1.61·11-s + 12-s − 4.89·13-s − 2.38·14-s + 0.380·15-s + 16-s − 3.09·17-s + 18-s − 5.09·19-s + 0.380·20-s − 2.38·21-s + 1.61·22-s − 23-s + 24-s − 4.85·25-s − 4.89·26-s + 27-s − 2.38·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.170·5-s + 0.408·6-s − 0.899·7-s + 0.353·8-s + 0.333·9-s + 0.120·10-s + 0.485·11-s + 0.288·12-s − 1.35·13-s − 0.636·14-s + 0.0982·15-s + 0.250·16-s − 0.750·17-s + 0.235·18-s − 1.16·19-s + 0.0851·20-s − 0.519·21-s + 0.343·22-s − 0.208·23-s + 0.204·24-s − 0.971·25-s − 0.960·26-s + 0.192·27-s − 0.449·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 0.380T + 5T^{2} \) |
| 7 | \( 1 + 2.38T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 + 3.09T + 17T^{2} \) |
| 19 | \( 1 + 5.09T + 19T^{2} \) |
| 31 | \( 1 + 8.59T + 31T^{2} \) |
| 37 | \( 1 - 1.94T + 37T^{2} \) |
| 41 | \( 1 + 1.80T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 8.12T + 47T^{2} \) |
| 53 | \( 1 + 5.95T + 53T^{2} \) |
| 59 | \( 1 - 8.88T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 0.712T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 0.574T + 73T^{2} \) |
| 79 | \( 1 + 3.86T + 79T^{2} \) |
| 83 | \( 1 + 9.03T + 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.033715765663999358077183265542, −7.08894138634777720334154716751, −6.71376943057184473451704042572, −5.86965716255215612637342423823, −5.00183405789406268817902343628, −4.10440551720196885775810887163, −3.54262857095101012713557881753, −2.47610424929641498032675819550, −1.92479423787572138315902367240, 0,
1.92479423787572138315902367240, 2.47610424929641498032675819550, 3.54262857095101012713557881753, 4.10440551720196885775810887163, 5.00183405789406268817902343628, 5.86965716255215612637342423823, 6.71376943057184473451704042572, 7.08894138634777720334154716751, 8.033715765663999358077183265542
