| L(s) = 1 | + (−1.58 − 0.707i)3-s − 1.41i·5-s + (−1 + 2.44i)7-s + (2.00 + 2.23i)9-s + 3.46·11-s − 3.16·13-s + (−1.00 + 2.23i)15-s + 3.16·19-s + (3.31 − 3.16i)21-s + 4.47i·23-s + 2.99·25-s + (−1.58 − 4.94i)27-s + 6.92·29-s − 4.89i·31-s + (−5.47 − 2.44i)33-s + ⋯ |
| L(s) = 1 | + (−0.912 − 0.408i)3-s − 0.632i·5-s + (−0.377 + 0.925i)7-s + (0.666 + 0.745i)9-s + 1.04·11-s − 0.877·13-s + (−0.258 + 0.577i)15-s + 0.725·19-s + (0.722 − 0.690i)21-s + 0.932i·23-s + 0.599·25-s + (−0.304 − 0.952i)27-s + 1.28·29-s − 0.879i·31-s + (−0.953 − 0.426i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09261 - 0.0815908i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09261 - 0.0815908i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.58 + 0.707i)T \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 7.74iT - 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 9.89iT - 59T^{2} \) |
| 61 | \( 1 + 3.16T + 61T^{2} \) |
| 67 | \( 1 - 7.74iT - 67T^{2} \) |
| 71 | \( 1 + 8.94iT - 71T^{2} \) |
| 73 | \( 1 - 14.6iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 7.07iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−10.52231907671740545539140093210, −9.514588175173553116692170127298, −8.949601816091038271357286677464, −7.72553818925245935301051168122, −6.86079774798308497203559389869, −5.89288934759302973317747365924, −5.21922746913941904566261370964, −4.16688667075670683940645121724, −2.50174007538239128358724425673, −1.02504051934004328024292856779,
0.919538965353800595283517205868, 2.98885799164017745486805745354, 4.12381235344667461555755829206, 4.91987906716378953559767913180, 6.29857623882630715409565390408, 6.77351062082377305922502348667, 7.61085626596294003261011501256, 9.112703495114881159857411499091, 9.856809400430012985443486776874, 10.58321227686992431808298708283
