L(s) = 1 | + (1.19 + 2.06i)2-s + 2.75·3-s + (−1.85 + 3.20i)4-s + (0.491 − 0.850i)5-s + (3.28 + 5.69i)6-s − 4.06·8-s + 4.57·9-s + 2.34·10-s − 0.587·11-s + (−5.09 + 8.82i)12-s + (−2.39 − 2.69i)13-s + (1.35 − 2.34i)15-s + (−1.15 − 1.99i)16-s + (−3.22 + 5.58i)17-s + (5.45 + 9.45i)18-s + 3.82·19-s + ⋯ |
L(s) = 1 | + (0.844 + 1.46i)2-s + 1.58·3-s + (−0.925 + 1.60i)4-s + (0.219 − 0.380i)5-s + (1.34 + 2.32i)6-s − 1.43·8-s + 1.52·9-s + 0.741·10-s − 0.177·11-s + (−1.47 + 2.54i)12-s + (−0.663 − 0.748i)13-s + (0.348 − 0.604i)15-s + (−0.288 − 0.498i)16-s + (−0.782 + 1.35i)17-s + (1.28 + 2.22i)18-s + 0.877·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.09578 + 2.93478i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.09578 + 2.93478i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (2.39 + 2.69i)T \) |
good | 2 | \( 1 + (-1.19 - 2.06i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 - 2.75T + 3T^{2} \) |
| 5 | \( 1 + (-0.491 + 0.850i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 0.587T + 11T^{2} \) |
| 17 | \( 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 + (4.13 + 7.15i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.98 + 3.42i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.49 + 2.58i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.877 + 1.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.83 + 3.17i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.19 + 5.52i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.17 - 3.75i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.212 + 0.368i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.00 + 5.20i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 2.20T + 61T^{2} \) |
| 67 | \( 1 - 7.01T + 67T^{2} \) |
| 71 | \( 1 + (1.80 + 3.11i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.46 - 4.27i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.39 - 2.41i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.86T + 83T^{2} \) |
| 89 | \( 1 + (1.04 + 1.81i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.84 - 6.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.57254271953480297893811384964, −9.578263712540077930859947670022, −8.619271177109991079698889772786, −8.137436262557387223355342975252, −7.43105518863267110894059917507, −6.43670166354613625656057457561, −5.38743016778553271261975237130, −4.37625520513206671135161555368, −3.51532467660400654985259482081, −2.25400567755813143034188057719,
1.70018893531514244891975146749, 2.64586874865612167233941198747, 3.24029792458838389282794079176, 4.30880335296372158585092852009, 5.21928711606582431499280948284, 6.86992702818170139894133386171, 7.76961000084045241619079786000, 9.031690461089117871164551586103, 9.614134440101704674980559633898, 10.20191663217072024872166787372