L(s) = 1 | + (0.156 − 0.987i)2-s + (−0.951 − 0.309i)4-s + (−0.280 − 0.550i)7-s + (−0.453 + 0.891i)8-s + (0.587 + 0.809i)9-s + (−0.309 + 0.951i)11-s + (1.59 + 0.253i)13-s + (−0.587 + 0.190i)14-s + (0.809 + 0.587i)16-s + (0.891 − 0.453i)18-s + (0.363 + 1.11i)19-s + (0.891 + 0.453i)22-s + (−1.34 + 1.34i)23-s + (0.5 − 1.53i)26-s + (0.0966 + 0.610i)28-s + ⋯ |
L(s) = 1 | + (0.156 − 0.987i)2-s + (−0.951 − 0.309i)4-s + (−0.280 − 0.550i)7-s + (−0.453 + 0.891i)8-s + (0.587 + 0.809i)9-s + (−0.309 + 0.951i)11-s + (1.59 + 0.253i)13-s + (−0.587 + 0.190i)14-s + (0.809 + 0.587i)16-s + (0.891 − 0.453i)18-s + (0.363 + 1.11i)19-s + (0.891 + 0.453i)22-s + (−1.34 + 1.34i)23-s + (0.5 − 1.53i)26-s + (0.0966 + 0.610i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.152115491\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.152115491\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.156 + 0.987i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (0.309 - 0.951i)T \) |
good | 3 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 + (0.280 + 0.550i)T + (-0.587 + 0.809i)T^{2} \) |
| 13 | \( 1 + (-1.59 - 0.253i)T + (0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 19 | \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (1.34 - 1.34i)T - iT^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-1.04 + 0.533i)T + (0.587 - 0.809i)T^{2} \) |
| 41 | \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.533 + 1.04i)T + (-0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (-0.297 + 1.87i)T + (-0.951 - 0.309i)T^{2} \) |
| 59 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 89 | \( 1 - 0.618iT - T^{2} \) |
| 97 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
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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
−9.518077649132806721064442561108, −8.395114099612429640959324233840, −7.86160409814727473599406707893, −6.89090006376065853032648259376, −5.80475511797025434205700237769, −5.07270162999177673574075033897, −3.88850847905997509708192749129, −3.73001320708557504630139335024, −2.13900383238298307393073853555, −1.42836479715285732557789488873,
0.879126985253945424015587541845, 2.84909919612799830037712047324, 3.71891079656527404163483033365, 4.50787345578506883387474044352, 5.68085832418690579706790838456, 6.16699536473097561399273445437, 6.75001025861789888413187346721, 7.78958051807034868506419766562, 8.623474679019507690900116536027, 8.907525370933474339120638034357