L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1 + 1.73i)5-s + 0.999·8-s + (1.5 − 2.59i)9-s + (−0.999 − 1.73i)10-s + (0.5 + 0.866i)11-s + 2·13-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)17-s + (1.5 + 2.59i)18-s + 1.99·20-s − 0.999·22-s + (4 − 6.92i)23-s + (0.500 + 0.866i)25-s + (−1 + 1.73i)26-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.447 + 0.774i)5-s + 0.353·8-s + (0.5 − 0.866i)9-s + (−0.316 − 0.547i)10-s + (0.150 + 0.261i)11-s + 0.554·13-s + (−0.125 + 0.216i)16-s + (−0.242 − 0.420i)17-s + (0.353 + 0.612i)18-s + 0.447·20-s − 0.213·22-s + (0.834 − 1.44i)23-s + (0.100 + 0.173i)25-s + (−0.196 + 0.339i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.277208502\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.277208502\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 3 | \( 1 + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.866680128154662263462566959207, −9.114758760581381904312874453900, −8.357984381276078225586116532786, −7.28250147827492115171558680778, −6.80634681628425936349539485406, −6.08806478095506394916109331882, −4.79551662724391612437295657993, −3.87708234342747849797420863336, −2.75724397740622672694641407672, −0.982740643493707664849252053046,
0.927084690643328753274381074031, 2.11151157087051574290647776856, 3.51839276428916538272347388440, 4.38641952433449779125522636403, 5.23118869012221395816040035325, 6.41479687663417399200876361941, 7.72415773165913018780646671439, 8.044698259121380828699230515823, 9.087765083059639597812593902775, 9.624130535084563857105116220844