| L(s) = 1 | − 2-s + 4-s + (−1.35 − 2.27i)7-s − 8-s + 2.71i·11-s + 6.54·13-s + (1.35 + 2.27i)14-s + 16-s − 1.53i·17-s + 2.30i·19-s − 2.71i·22-s + 3.83·23-s − 6.54·26-s + (−1.35 − 2.27i)28-s + 3.83i·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.512 − 0.858i)7-s − 0.353·8-s + 0.817i·11-s + 1.81·13-s + (0.362 + 0.607i)14-s + 0.250·16-s − 0.371i·17-s + 0.528i·19-s − 0.577i·22-s + 0.799·23-s − 1.28·26-s + (−0.256 − 0.429i)28-s + 0.711i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.311251006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.311251006\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (1.35 + 2.27i)T \) |
| good | 11 | \( 1 - 2.71iT - 11T^{2} \) |
| 13 | \( 1 - 6.54T + 13T^{2} \) |
| 17 | \( 1 + 1.53iT - 17T^{2} \) |
| 19 | \( 1 - 2.30iT - 19T^{2} \) |
| 23 | \( 1 - 3.83T + 23T^{2} \) |
| 29 | \( 1 - 3.83iT - 29T^{2} \) |
| 31 | \( 1 + 3.25iT - 31T^{2} \) |
| 37 | \( 1 + 3.01iT - 37T^{2} \) |
| 41 | \( 1 + 6.54T + 41T^{2} \) |
| 43 | \( 1 + 0.468iT - 43T^{2} \) |
| 47 | \( 1 - 9.11iT - 47T^{2} \) |
| 53 | \( 1 - 9.25T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 4.78iT - 61T^{2} \) |
| 67 | \( 1 + 13.5iT - 67T^{2} \) |
| 71 | \( 1 - 2.30iT - 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 13.0iT - 83T^{2} \) |
| 89 | \( 1 - 9.60T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−8.836186323373190427508801527678, −7.87951487224000898044930965757, −7.30287894964034613796965221190, −6.54500315852757201891308577131, −5.93901946020542669981545952318, −4.77664936211991705010310734024, −3.80968647285223689358161778168, −3.12560491307407065571191377562, −1.76655945762699630905003130484, −0.841240448747767722625190576284,
0.74364155357605945851176613420, 1.90439262616827055312852490495, 3.08271248630094405356581953395, 3.61712103918741698915553791039, 4.98944271761591133291664418826, 5.99486650216455152743862999009, 6.28512127082772124579441920337, 7.19497801200169312405703086893, 8.262523271792886237722205955151, 8.774219561015748738888609065725
