| L(s) = 1 | + i·2-s + 3-s − 4-s − i·5-s + i·6-s + 3.64i·7-s − i·8-s + 9-s + 10-s − 1.66i·11-s − 12-s − 3.64·14-s − i·15-s + 16-s − 4·17-s + i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s + 1.37i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s − 0.502i·11-s − 0.288·12-s − 0.974·14-s − 0.258i·15-s + 0.250·16-s − 0.970·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.014344695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.014344695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 3.64iT - 7T^{2} \) |
| 11 | \( 1 + 1.66iT - 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 6.31iT - 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 - 4.21iT - 31T^{2} \) |
| 37 | \( 1 - 9.86iT - 37T^{2} \) |
| 41 | \( 1 - 9.28iT - 41T^{2} \) |
| 43 | \( 1 - 7.57T + 43T^{2} \) |
| 47 | \( 1 + 6.82iT - 47T^{2} \) |
| 53 | \( 1 + 0.848T + 53T^{2} \) |
| 59 | \( 1 + 6.10iT - 59T^{2} \) |
| 61 | \( 1 - 7.46T + 61T^{2} \) |
| 67 | \( 1 - 14.7iT - 67T^{2} \) |
| 71 | \( 1 - 3.51iT - 71T^{2} \) |
| 73 | \( 1 - 12.2iT - 73T^{2} \) |
| 79 | \( 1 - 9.93T + 79T^{2} \) |
| 83 | \( 1 - 7.95iT - 83T^{2} \) |
| 89 | \( 1 - 5.95iT - 89T^{2} \) |
| 97 | \( 1 - 2.75iT - 97T^{2} \) |
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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
−8.506799408552134893183959697858, −8.017652614832919492164572742672, −6.72062428750915502874707952079, −6.56173611980093276972433107993, −5.45597634583478640079345190630, −4.91573590721944879895350201584, −4.18148761478952584878192897097, −2.93717393313180031438687823815, −2.43592536100199499628124598762, −1.02914151714008510796555216932,
0.57230767564293508799096106094, 1.76912571929869992524067896924, 2.51344754011436646194025510701, 3.56946332700340656582377528623, 4.08744570549046923557424715834, 4.66691390996697603577103341827, 5.91565605009256711749291415892, 6.69850219500873487898547462761, 7.54078455149345156818020175414, 7.86886358115757632517688073386
