| L(s) = 1 | + (−0.461 + 1.66i)3-s − 1.99i·5-s + 1.52i·7-s + (−2.57 − 1.54i)9-s − 2.82·11-s + 13-s + (3.32 + 0.920i)15-s − 4.93i·17-s − 0.312i·19-s + (−2.55 − 0.705i)21-s + 0.713·23-s + 1.03·25-s + (3.76 − 3.58i)27-s − 2.82i·29-s − 1.32i·31-s + ⋯ |
| L(s) = 1 | + (−0.266 + 0.963i)3-s − 0.890i·5-s + 0.577i·7-s + (−0.857 − 0.514i)9-s − 0.852·11-s + 0.277·13-s + (0.858 + 0.237i)15-s − 1.19i·17-s − 0.0716i·19-s + (−0.556 − 0.153i)21-s + 0.148·23-s + 0.206·25-s + (0.724 − 0.689i)27-s − 0.525i·29-s − 0.237i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9878849222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9878849222\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.461 - 1.66i)T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 1.99iT - 5T^{2} \) |
| 7 | \( 1 - 1.52iT - 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 17 | \( 1 + 4.93iT - 17T^{2} \) |
| 19 | \( 1 + 0.312iT - 19T^{2} \) |
| 23 | \( 1 - 0.713T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 1.32iT - 31T^{2} \) |
| 37 | \( 1 + 6.18T + 37T^{2} \) |
| 41 | \( 1 + 7.19iT - 41T^{2} \) |
| 43 | \( 1 + 9.95iT - 43T^{2} \) |
| 47 | \( 1 - 4.97T + 47T^{2} \) |
| 53 | \( 1 + 0.271iT - 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 8.31iT - 67T^{2} \) |
| 71 | \( 1 - 7.19T + 71T^{2} \) |
| 73 | \( 1 - 4.43T + 73T^{2} \) |
| 79 | \( 1 + 5.21iT - 79T^{2} \) |
| 83 | \( 1 - 1.49T + 83T^{2} \) |
| 89 | \( 1 - 4.43iT - 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−9.365473000490586135968559942843, −8.979797366692691552383838594032, −8.201834307225853528377165288084, −7.11438337702295121015515917823, −5.84978459305102228985922356276, −5.23305375898339260070034500342, −4.62336226648965706130700363823, −3.47657646280387222664970954928, −2.36617571902876304695679452438, −0.45283529082178779209392870951,
1.32065045037009087833595754875, 2.57767614251511629190130007719, 3.49341115651440527161792634730, 4.83987433593637954749951054450, 5.92084348402227114722930597774, 6.57861369227977220275573272048, 7.35061559912983944466253022680, 7.988135215108792947213824255163, 8.811497986345364064032814365137, 10.14212362105798156151452338736
