| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.282525510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.282525510\) |
| \(L(\frac{3}{2})\) |
|
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
−8.241066017164918423326273064085, −7.925269360330472723539009191616, −6.57365115083941343700062612609, −6.10165518070569603939440822247, −5.44206281756512148105080063140, −4.59349455321630967620455727030, −3.73185163875847412538437801830, −3.37648837836123856064318055157, −1.76214433108730324226448404186, −1.15042723026884830462839273726,
0.826615603579187692757830399967, 2.33652694715314080242597396020, 2.90173902508614706905702174047, 3.81841817558198060106123670048, 4.47016763459969971346736199079, 5.46474959810131070437984197229, 6.13882767208299922270009043614, 6.93182190057343728849099344178, 7.31920764367402418936798067920, 8.171260970561351304319662222746
