| L(s) = 1 | + 3.46i·5-s + i·7-s + 17.3·11-s − 1.73i·13-s + 6·17-s + 1.73·19-s − 30i·23-s + 13.0·25-s − 20.7i·29-s + 14i·31-s − 3.46·35-s − 19.0i·37-s − 48·41-s + 20.7·43-s + 66i·47-s + ⋯ |
| L(s) = 1 | + 0.692i·5-s + 0.142i·7-s + 1.57·11-s − 0.133i·13-s + 0.352·17-s + 0.0911·19-s − 1.30i·23-s + 0.520·25-s − 0.716i·29-s + 0.451i·31-s − 0.0989·35-s − 0.514i·37-s − 1.17·41-s + 0.483·43-s + 1.40i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.323651008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.323651008\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.46iT - 25T^{2} \) |
| 7 | \( 1 - iT - 49T^{2} \) |
| 11 | \( 1 - 17.3T + 121T^{2} \) |
| 13 | \( 1 + 1.73iT - 169T^{2} \) |
| 17 | \( 1 - 6T + 289T^{2} \) |
| 19 | \( 1 - 1.73T + 361T^{2} \) |
| 23 | \( 1 + 30iT - 529T^{2} \) |
| 29 | \( 1 + 20.7iT - 841T^{2} \) |
| 31 | \( 1 - 14iT - 961T^{2} \) |
| 37 | \( 1 + 19.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 48T + 1.68e3T^{2} \) |
| 43 | \( 1 - 20.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 66iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 48.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 31.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 43.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 60.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 48iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 49T + 5.32e3T^{2} \) |
| 79 | \( 1 - 83iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 13.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 66T + 7.92e3T^{2} \) |
| 97 | \( 1 - 107T + 9.40e3T^{2} \) |
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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
−9.160150900250853417336530499515, −8.465025210964029240660731276263, −7.50571804425440727526269050749, −6.61225326912235546332118603996, −6.24561322722313000841246326000, −5.04398864026024653551730995206, −4.05991413309641783541491976341, −3.23114777176107155036790157718, −2.16284912108941812069817304217, −0.864995184155536630805374949012,
0.908430742446098265536796631963, 1.74670206546397509577829715902, 3.30065820220666940034144252849, 4.06417002691866182933559828920, 4.98905928838128514454043273786, 5.83316525517568769476645591477, 6.77138829857945738529493151900, 7.44719776587910512885330379130, 8.523128537141455728296351556891, 9.057731134062094270881969931368
