| L(s) = 1 | + i·3-s − i·4-s + 7-s − 9-s + 12-s + i·13-s − 16-s + (−1 − i)19-s + i·21-s + i·25-s − i·27-s − i·28-s + (−1 − i)31-s + i·36-s + (−1 − i)37-s + ⋯ |
| L(s) = 1 | + i·3-s − i·4-s + 7-s − 9-s + 12-s + i·13-s − 16-s + (−1 − i)19-s + i·21-s + i·25-s − i·27-s − i·28-s + (−1 − i)31-s + i·36-s + (−1 − i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7794359659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7794359659\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - iT \) |
| good | 2 | \( 1 + iT^{2} \) |
| 5 | \( 1 - iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (1 + i)T + iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (1 + i)T + iT^{2} \) |
| 37 | \( 1 + (1 + i)T + iT^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 - 2iT - T^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (-1 - i)T + iT^{2} \) |
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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
−11.70819651357079358850546289319, −11.08081182563556645174937686744, −10.44374118569693185526636673346, −9.250621674052433742983490598006, −8.791092690411252611951459921484, −7.22192787859305902232928466151, −5.89624474268842790965700946884, −4.96109938550202039274255284076, −4.11097336584119096265799655388, −2.09530318620797307391385453868,
2.00358764672848900001252726229, 3.42111817936412082676326102628, 4.95435600113777969347995469050, 6.28663700473568465667282707887, 7.42134746393355456449742691462, 8.179493264013941286077870283199, 8.682560625708748931078392308669, 10.45315970971979323276855676167, 11.36630232080213877764079627773, 12.34338953988064875969777253967
