L(s) = 1 | + (−0.850 − 0.753i)2-s + (0.0350 + 0.288i)4-s + (−0.748 − 0.663i)7-s + (−0.457 + 0.663i)8-s + (0.885 − 0.464i)9-s + (−0.627 − 1.65i)11-s + (0.136 + 1.12i)14-s + (1.17 − 0.288i)16-s + (−1.10 − 0.271i)18-s + (−0.713 + 1.88i)22-s − 0.709·23-s + (0.568 − 0.822i)25-s + (0.165 − 0.239i)28-s + (−0.402 + 1.06i)29-s + (−0.5 − 0.262i)32-s + ⋯ |
L(s) = 1 | + (−0.850 − 0.753i)2-s + (0.0350 + 0.288i)4-s + (−0.748 − 0.663i)7-s + (−0.457 + 0.663i)8-s + (0.885 − 0.464i)9-s + (−0.627 − 1.65i)11-s + (0.136 + 1.12i)14-s + (1.17 − 0.288i)16-s + (−1.10 − 0.271i)18-s + (−0.713 + 1.88i)22-s − 0.709·23-s + (0.568 − 0.822i)25-s + (0.165 − 0.239i)28-s + (−0.402 + 1.06i)29-s + (−0.5 − 0.262i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4714632134\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4714632134\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.748 + 0.663i)T \) |
| 53 | \( 1 + (-0.885 - 0.464i)T \) |
good | 2 | \( 1 + (0.850 + 0.753i)T + (0.120 + 0.992i)T^{2} \) |
| 3 | \( 1 + (-0.885 + 0.464i)T^{2} \) |
| 5 | \( 1 + (-0.568 + 0.822i)T^{2} \) |
| 11 | \( 1 + (0.627 + 1.65i)T + (-0.748 + 0.663i)T^{2} \) |
| 13 | \( 1 + (0.970 + 0.239i)T^{2} \) |
| 17 | \( 1 + (0.354 - 0.935i)T^{2} \) |
| 19 | \( 1 + (0.970 + 0.239i)T^{2} \) |
| 23 | \( 1 + 0.709T + T^{2} \) |
| 29 | \( 1 + (0.402 - 1.06i)T + (-0.748 - 0.663i)T^{2} \) |
| 31 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 37 | \( 1 + (0.234 - 0.0576i)T + (0.885 - 0.464i)T^{2} \) |
| 41 | \( 1 + (0.748 - 0.663i)T^{2} \) |
| 43 | \( 1 + (-1.88 - 0.464i)T + (0.885 + 0.464i)T^{2} \) |
| 47 | \( 1 + (-0.568 - 0.822i)T^{2} \) |
| 59 | \( 1 + (-0.568 - 0.822i)T^{2} \) |
| 61 | \( 1 + (0.354 + 0.935i)T^{2} \) |
| 67 | \( 1 + (-0.213 - 1.75i)T + (-0.970 + 0.239i)T^{2} \) |
| 71 | \( 1 + (-1.45 - 0.358i)T + (0.885 + 0.464i)T^{2} \) |
| 73 | \( 1 + (0.354 - 0.935i)T^{2} \) |
| 79 | \( 1 + (-1.12 + 0.992i)T + (0.120 - 0.992i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.354 - 0.935i)T^{2} \) |
| 97 | \( 1 + (-0.568 + 0.822i)T^{2} \) |
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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
−10.85622434075686611946121301664, −10.51456790047545035820039117775, −9.595143925151886733493227163620, −8.794745530563313412811667021767, −7.80886727686802541360030848074, −6.58569654058103126591125770165, −5.57451031639443130199152633790, −3.90387694008696335764591500808, −2.78481384888378661513213283484, −0.923004033153133149579871884429,
2.26978230391207159683779790934, 3.96906498058063653611713562258, 5.32225999196743651554559250781, 6.57371176271318143103679595360, 7.34942250951774511356334248066, 8.033155475641191286886712562493, 9.332723755707972582072247467829, 9.719556903973332351539893052502, 10.62404054658992641850037621278, 12.24059554237064927500102252741