L(s) = 1 | + (−0.321 − 1.37i)2-s + (−1.79 + 0.884i)4-s + (2.11 − 0.726i)5-s + 4.05i·7-s + (1.79 + 2.18i)8-s + (−1.67 − 2.67i)10-s − 0.985i·11-s + 4.94·13-s + (5.58 − 1.30i)14-s + (2.43 − 3.17i)16-s + 4.52i·17-s + 2.60i·19-s + (−3.15 + 3.17i)20-s + (−1.35 + 0.316i)22-s − 3.53i·23-s + ⋯ |
L(s) = 1 | + (−0.227 − 0.973i)2-s + (−0.896 + 0.442i)4-s + (0.945 − 0.324i)5-s + 1.53i·7-s + (0.634 + 0.773i)8-s + (−0.530 − 0.847i)10-s − 0.297i·11-s + 1.37·13-s + (1.49 − 0.348i)14-s + (0.608 − 0.793i)16-s + 1.09i·17-s + 0.597i·19-s + (−0.704 + 0.709i)20-s + (−0.289 + 0.0674i)22-s − 0.737i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27715 - 0.362475i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27715 - 0.362475i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.321 + 1.37i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.11 + 0.726i)T \) |
good | 7 | \( 1 - 4.05iT - 7T^{2} \) |
| 11 | \( 1 + 0.985iT - 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 - 4.52iT - 17T^{2} \) |
| 19 | \( 1 - 2.60iT - 19T^{2} \) |
| 23 | \( 1 + 3.53iT - 23T^{2} \) |
| 29 | \( 1 + 7.59iT - 29T^{2} \) |
| 31 | \( 1 + 3.28T + 31T^{2} \) |
| 37 | \( 1 + 0.945T + 37T^{2} \) |
| 41 | \( 1 + 0.568T + 41T^{2} \) |
| 43 | \( 1 - 8.45T + 43T^{2} \) |
| 47 | \( 1 - 2.60iT - 47T^{2} \) |
| 53 | \( 1 + 0.229T + 53T^{2} \) |
| 59 | \( 1 - 9.10iT - 59T^{2} \) |
| 61 | \( 1 + 11.0iT - 61T^{2} \) |
| 67 | \( 1 + 8.45T + 67T^{2} \) |
| 71 | \( 1 + 1.43T + 71T^{2} \) |
| 73 | \( 1 - 11.9iT - 73T^{2} \) |
| 79 | \( 1 - 3.28T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 3.23iT - 97T^{2} \) |
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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
−11.30708823164219970528500441346, −10.48253364582515749924723852874, −9.507210250935130341607667125435, −8.721779281457948772792267462891, −8.231089104083700738185852813907, −6.10015587382555690239644792156, −5.60811020835615046518081378886, −4.11484246978006869203802557296, −2.68714512932063640124658303348, −1.60176374548220699099568604134,
1.22089145497875338021484942545, 3.53152812415233582771992761570, 4.78502043571790355248690623310, 5.87461014716662849957506604185, 6.94722642711270547252406680433, 7.39184532204486221626809477867, 8.794918425839892856557094068140, 9.561935837335683079899355336616, 10.49025458496530421932198093420, 11.08913878921240049237990474693