| L(s) = 1 | + 1.41i·3-s − 3.16·7-s + 0.999·9-s − 4.47i·21-s + 9.48·23-s + 5.65i·27-s − 8.94i·29-s + 12·41-s + 12.7i·43-s − 9.48·47-s + 3.00·49-s + 13.4i·61-s − 3.16·63-s − 4.24i·67-s + 13.4i·69-s + ⋯ |
| L(s) = 1 | + 0.816i·3-s − 1.19·7-s + 0.333·9-s − 0.975i·21-s + 1.97·23-s + 1.08i·27-s − 1.66i·29-s + 1.87·41-s + 1.94i·43-s − 1.38·47-s + 0.428·49-s + 1.71i·61-s − 0.398·63-s − 0.518i·67-s + 1.61i·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.528098245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.528098245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.41iT - 3T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 9.48T + 23T^{2} \) |
| 29 | \( 1 + 8.94iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 12.7iT - 43T^{2} \) |
| 47 | \( 1 + 9.48T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 13.4iT - 61T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 15.5iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.267734746584264247516021152531, −8.145784026876213278835904311687, −7.29132315930635251732461621340, −6.56369403976058871638616933801, −5.86197909919710251196310555134, −4.85704260264121078331203349494, −4.20799995469679379846350094327, −3.32575124432422205203599727838, −2.61432698199595849931782372763, −1.00821793580506145359155993651,
0.57913438628799109247460072118, 1.68515344867657993320041582016, 2.85244636023037056017606456719, 3.53110413824044632371810507149, 4.65937574001803619265214304332, 5.55258207921656896071548240671, 6.48799290492837190035818987181, 6.96350641164489080670922143511, 7.47046846487800291775187937119, 8.515398992535737671923890554448
