| L(s) = 1 | − 2.17·2-s + 2.72·4-s + 1.26·5-s + 1.97·7-s − 1.58·8-s − 2.74·10-s + 2.80·11-s − 4.91·13-s − 4.29·14-s − 2.01·16-s + 7.61·17-s + 0.948·19-s + 3.44·20-s − 6.10·22-s − 23-s − 3.41·25-s + 10.6·26-s + 5.38·28-s − 29-s + 0.780·31-s + 7.54·32-s − 16.5·34-s + 2.48·35-s + 2.59·37-s − 2.06·38-s − 1.99·40-s − 1.16·41-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.36·4-s + 0.563·5-s + 0.745·7-s − 0.560·8-s − 0.866·10-s + 0.845·11-s − 1.36·13-s − 1.14·14-s − 0.502·16-s + 1.84·17-s + 0.217·19-s + 0.769·20-s − 1.30·22-s − 0.208·23-s − 0.682·25-s + 2.09·26-s + 1.01·28-s − 0.185·29-s + 0.140·31-s + 1.33·32-s − 2.84·34-s + 0.420·35-s + 0.426·37-s − 0.334·38-s − 0.315·40-s − 0.181·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.160252967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160252967\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 5 | \( 1 - 1.26T + 5T^{2} \) |
| 7 | \( 1 - 1.97T + 7T^{2} \) |
| 11 | \( 1 - 2.80T + 11T^{2} \) |
| 13 | \( 1 + 4.91T + 13T^{2} \) |
| 17 | \( 1 - 7.61T + 17T^{2} \) |
| 19 | \( 1 - 0.948T + 19T^{2} \) |
| 31 | \( 1 - 0.780T + 31T^{2} \) |
| 37 | \( 1 - 2.59T + 37T^{2} \) |
| 41 | \( 1 + 1.16T + 41T^{2} \) |
| 43 | \( 1 + 3.06T + 43T^{2} \) |
| 47 | \( 1 + 5.32T + 47T^{2} \) |
| 53 | \( 1 - 9.00T + 53T^{2} \) |
| 59 | \( 1 - 8.64T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 2.24T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 1.58T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 9.44T + 89T^{2} \) |
| 97 | \( 1 + 8.45T + 97T^{2} \) |
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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
−8.148316421578951827000565860861, −7.50828901128932909925232733724, −7.06972364058961145019238926168, −6.09368589393638267727837138649, −5.33895137912347806918573536302, −4.55599125020205422335893795851, −3.43688563124162868483524288364, −2.29230240884898125424820339851, −1.63588504466586665174512503449, −0.75324726970035300658404284586,
0.75324726970035300658404284586, 1.63588504466586665174512503449, 2.29230240884898125424820339851, 3.43688563124162868483524288364, 4.55599125020205422335893795851, 5.33895137912347806918573536302, 6.09368589393638267727837138649, 7.06972364058961145019238926168, 7.50828901128932909925232733724, 8.148316421578951827000565860861
