| L(s) = 1 | + (1.20 + 0.738i)2-s + (0.681 − 0.317i)3-s + (0.909 + 1.78i)4-s + (−2.29 + 3.27i)5-s + (1.05 + 0.119i)6-s + (2.44 − 4.23i)7-s + (−0.217 + 2.82i)8-s + (−1.56 + 1.86i)9-s + (−5.18 + 2.25i)10-s + (1.70 + 0.456i)11-s + (1.18 + 0.924i)12-s + (3.65 + 1.70i)13-s + (6.07 − 3.30i)14-s + (−0.521 + 2.95i)15-s + (−2.34 + 3.24i)16-s + (−1.14 + 0.960i)17-s + ⋯ |
| L(s) = 1 | + (0.852 + 0.522i)2-s + (0.393 − 0.183i)3-s + (0.454 + 0.890i)4-s + (−1.02 + 1.46i)5-s + (0.431 + 0.0489i)6-s + (0.924 − 1.60i)7-s + (−0.0769 + 0.997i)8-s + (−0.521 + 0.621i)9-s + (−1.63 + 0.713i)10-s + (0.514 + 0.137i)11-s + (0.342 + 0.266i)12-s + (1.01 + 0.472i)13-s + (1.62 − 0.882i)14-s + (−0.134 + 0.763i)15-s + (−0.586 + 0.810i)16-s + (−0.277 + 0.233i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67642 + 1.31283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67642 + 1.31283i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.20 - 0.738i)T \) |
| 19 | \( 1 + (3.55 + 2.52i)T \) |
| good | 3 | \( 1 + (-0.681 + 0.317i)T + (1.92 - 2.29i)T^{2} \) |
| 5 | \( 1 + (2.29 - 3.27i)T + (-1.71 - 4.69i)T^{2} \) |
| 7 | \( 1 + (-2.44 + 4.23i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.70 - 0.456i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-3.65 - 1.70i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (1.14 - 0.960i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.963 + 5.46i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.181 + 2.07i)T + (-28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (-2.78 + 4.81i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.74 + 5.74i)T - 37iT^{2} \) |
| 41 | \( 1 + (-3.28 - 1.19i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (2.02 + 1.41i)T + (14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-0.704 + 0.839i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.04 - 1.48i)T + (-18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (5.93 + 0.518i)T + (58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (-5.41 - 7.72i)T + (-20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (1.91 - 0.167i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (2.23 - 0.393i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (2.79 - 7.67i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (3.50 + 1.27i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-8.01 + 2.14i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (12.5 - 4.55i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-11.9 - 14.1i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−11.69550208265265056076040560451, −11.00523213891024489053554455684, −10.72091828101064227672596476557, −8.502937607867119886472622479228, −7.80879227575761081113251640580, −7.08051254221014314722008161720, −6.32912677097606793435115395271, −4.35531840900360845096615104934, −3.92414236619900375649967407919, −2.50455716431828799291598887684,
1.43659820785576536109877760337, 3.19289587527908771317126444551, 4.29138831352646982444927244949, 5.26734069653768524713614253347, 6.11801187254968466945066212851, 8.101163051506439590132487149027, 8.715338961375743043533858071632, 9.374575080734663358229863997294, 11.15312064630680206391132444416, 11.73615610701410510263474502043
