| L(s) = 1 | − i·3-s − 1.12·5-s + (2.11 − 1.59i)7-s − 9-s − 5.11·11-s + 5.88·13-s + 1.12i·15-s − 3.31i·17-s − 7.49i·19-s + (−1.59 − 2.11i)21-s + 1.73i·23-s − 3.72·25-s + i·27-s − 5.88i·29-s − 6.04·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.504·5-s + (0.798 − 0.601i)7-s − 0.333·9-s − 1.54·11-s + 1.63·13-s + 0.291i·15-s − 0.804i·17-s − 1.71i·19-s + (−0.347 − 0.461i)21-s + 0.362i·23-s − 0.745·25-s + 0.192i·27-s − 1.09i·29-s − 1.08·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.662890 - 0.978847i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.662890 - 0.978847i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (-2.11 + 1.59i)T \) |
| good | 5 | \( 1 + 1.12T + 5T^{2} \) |
| 11 | \( 1 + 5.11T + 11T^{2} \) |
| 13 | \( 1 - 5.88T + 13T^{2} \) |
| 17 | \( 1 + 3.31iT - 17T^{2} \) |
| 19 | \( 1 + 7.49iT - 19T^{2} \) |
| 23 | \( 1 - 1.73iT - 23T^{2} \) |
| 29 | \( 1 + 5.88iT - 29T^{2} \) |
| 31 | \( 1 + 6.04T + 31T^{2} \) |
| 37 | \( 1 - 1.65iT - 37T^{2} \) |
| 41 | \( 1 + 1.45iT - 41T^{2} \) |
| 43 | \( 1 + 1.79T + 43T^{2} \) |
| 47 | \( 1 - 5.56T + 47T^{2} \) |
| 53 | \( 1 + 3.62iT - 53T^{2} \) |
| 59 | \( 1 - 0.767iT - 59T^{2} \) |
| 61 | \( 1 - 0.317T + 61T^{2} \) |
| 67 | \( 1 - 6.56T + 67T^{2} \) |
| 71 | \( 1 + 10.1iT - 71T^{2} \) |
| 73 | \( 1 + 6.63iT - 73T^{2} \) |
| 79 | \( 1 - 3.01iT - 79T^{2} \) |
| 83 | \( 1 - 16.0iT - 83T^{2} \) |
| 89 | \( 1 - 8.08iT - 89T^{2} \) |
| 97 | \( 1 + 0.357iT - 97T^{2} \) |
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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
−10.52453559658813001568865453423, −9.227195988022990712883933751528, −8.236010416678153207774390960950, −7.69710497444849548286874888379, −6.91097026007351426861683936984, −5.66876397945138497591439806322, −4.76071790329417869411018728172, −3.56897707024067349215608494345, −2.25048403098704289405131093200, −0.63691144179567192327714280742,
1.80851988537845842965042136436, 3.33919777627592612222570279025, 4.21185366698261103863470090915, 5.45501246708668221427638501106, 5.94669334706951807595438299061, 7.58816371048313425895834492135, 8.288796926327309009860122735013, 8.785261362175048787286606624518, 10.15549055056266280921323118779, 10.76525758702197968171925311175
