| L(s) = 1 | − 2-s + 4-s + (−0.426 − 2.19i)5-s − 8-s + (0.426 + 2.19i)10-s + 1.28i·11-s − 6.14·13-s + 16-s + 6.52i·17-s − 6.03i·19-s + (−0.426 − 2.19i)20-s − 1.28i·22-s − 2.87·23-s + (−4.63 + 1.87i)25-s + 6.14·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.190 − 0.981i)5-s − 0.353·8-s + (0.134 + 0.694i)10-s + 0.386i·11-s − 1.70·13-s + 0.250·16-s + 1.58i·17-s − 1.38i·19-s + (−0.0953 − 0.490i)20-s − 0.273i·22-s − 0.600·23-s + (−0.927 + 0.374i)25-s + 1.20·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8242252766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8242252766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.426 + 2.19i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 1.28iT - 11T^{2} \) |
| 13 | \( 1 + 6.14T + 13T^{2} \) |
| 17 | \( 1 - 6.52iT - 17T^{2} \) |
| 19 | \( 1 + 6.03iT - 19T^{2} \) |
| 23 | \( 1 + 2.87T + 23T^{2} \) |
| 29 | \( 1 - 1.35iT - 29T^{2} \) |
| 31 | \( 1 + 8.65iT - 31T^{2} \) |
| 37 | \( 1 - 9.53iT - 37T^{2} \) |
| 41 | \( 1 - 8.71T + 41T^{2} \) |
| 43 | \( 1 + 5.35iT - 43T^{2} \) |
| 47 | \( 1 - 0.806iT - 47T^{2} \) |
| 53 | \( 1 - 6.66T + 53T^{2} \) |
| 59 | \( 1 - 1.59T + 59T^{2} \) |
| 61 | \( 1 - 6.35iT - 61T^{2} \) |
| 67 | \( 1 + 5.39iT - 67T^{2} \) |
| 71 | \( 1 - 15.6iT - 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 3.74iT - 83T^{2} \) |
| 89 | \( 1 + 3.63T + 89T^{2} \) |
| 97 | \( 1 - 8.76T + 97T^{2} \) |
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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
−8.466962338614858912643439956099, −7.70268151914277647400244868818, −7.22807136656248397629522398601, −6.26469582147244709038732130106, −5.45547790972767327433270453023, −4.62084847746325882555658589483, −4.00676062768648688833983535918, −2.63121517417007484139723898652, −1.90889492896928326687120216250, −0.69146989136365419914772884530,
0.43185993766307105973892399910, 2.00966075064107086512145388350, 2.72930740668284285293678132969, 3.49849045870161234432755014446, 4.58805033180588993207143777730, 5.54654716900104643206285509203, 6.27348469458225378379894548137, 7.26315083118567363604361619362, 7.41680009166326489128153484039, 8.179450199051084568665907902432
