| L(s) = 1 | − i·2-s + 3-s − 4-s + i·5-s − i·6-s + 4.64i·7-s + i·8-s + 9-s + 10-s − 4.40i·11-s − 12-s + 4.64·14-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 + 1.75i·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s − 1.32i·11-s − 0.288·12-s + 1.24·14-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
| \(L(1)\) |
\(\approx\) |
\(1.595124642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.595124642\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.64iT - 7T^{2} \) |
| 11 | \( 1 + 4.40iT - 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 8.04iT - 19T^{2} \) |
| 23 | \( 1 - 0.976T + 23T^{2} \) |
| 29 | \( 1 + 4.31T + 29T^{2} \) |
| 31 | \( 1 + 6.44iT - 31T^{2} \) |
| 37 | \( 1 + 3.79iT - 37T^{2} \) |
| 41 | \( 1 - 7.28iT - 41T^{2} \) |
| 43 | \( 1 + 0.716T + 43T^{2} \) |
| 47 | \( 1 + 9.75iT - 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 + 2.18iT - 59T^{2} \) |
| 61 | \( 1 - 7.46T + 61T^{2} \) |
| 67 | \( 1 - 1.82iT - 67T^{2} \) |
| 71 | \( 1 + 7.95iT - 71T^{2} \) |
| 73 | \( 1 - 4.36iT - 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 3.51iT - 83T^{2} \) |
| 89 | \( 1 + 8.17iT - 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−8.462822524450322271660733521551, −7.39027949892000718302803375410, −6.50553090572348648291693869126, −5.75637514519122680755566287765, −5.09802833306828247161283756880, −4.09054187225916811252795486847, −3.15240184070103368826777024259, −2.56982898737060044489727010172, −2.04578680122346428791950431993, −0.41356177095274225491966408421,
1.12060145097180172342011322828, 1.99075465261523000778943369703, 3.47920361038353043202764157459, 4.14888317682932836177645133613, 4.56200807356910771809942934681, 5.51233102816090678997540910540, 6.55285542337281023928162282602, 7.24233613279817751668776797624, 7.50305926153854632659366596970, 8.340117834616748919185064237907
