| L(s) = 1 | + 1.07i·2-s + 0.846·4-s − 1.27i·5-s + 1.02i·7-s + 3.05i·8-s + 1.37·10-s + 4.66i·11-s + (1.25 + 3.37i)13-s − 1.10·14-s − 1.58·16-s − 0.476·17-s − 6.69i·19-s − 1.08i·20-s − 5.00·22-s + 0.959·23-s + ⋯ |
| L(s) = 1 | + 0.759i·2-s + 0.423·4-s − 0.570i·5-s + 0.388i·7-s + 1.08i·8-s + 0.433·10-s + 1.40i·11-s + (0.348 + 0.937i)13-s − 0.295·14-s − 0.397·16-s − 0.115·17-s − 1.53i·19-s − 0.241i·20-s − 1.06·22-s + 0.200·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.804342099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804342099\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + (-1.25 - 3.37i)T \) |
| good | 2 | \( 1 - 1.07iT - 2T^{2} \) |
| 5 | \( 1 + 1.27iT - 5T^{2} \) |
| 7 | \( 1 - 1.02iT - 7T^{2} \) |
| 11 | \( 1 - 4.66iT - 11T^{2} \) |
| 17 | \( 1 + 0.476T + 17T^{2} \) |
| 19 | \( 1 + 6.69iT - 19T^{2} \) |
| 23 | \( 1 - 0.959T + 23T^{2} \) |
| 29 | \( 1 + 9.37T + 29T^{2} \) |
| 31 | \( 1 - 1.92iT - 31T^{2} \) |
| 37 | \( 1 - 4.94iT - 37T^{2} \) |
| 41 | \( 1 - 1.52iT - 41T^{2} \) |
| 43 | \( 1 - 2.62T + 43T^{2} \) |
| 47 | \( 1 - 6.83iT - 47T^{2} \) |
| 53 | \( 1 - 0.582T + 53T^{2} \) |
| 59 | \( 1 - 4.21iT - 59T^{2} \) |
| 61 | \( 1 - 9.43T + 61T^{2} \) |
| 67 | \( 1 - 2.32iT - 67T^{2} \) |
| 71 | \( 1 - 1.35iT - 71T^{2} \) |
| 73 | \( 1 - 12.8iT - 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 10.2iT - 83T^{2} \) |
| 89 | \( 1 + 6.85iT - 89T^{2} \) |
| 97 | \( 1 + 17.2iT - 97T^{2} \) |
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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
−9.993090546432539000101678047706, −9.045719857148376692291844403933, −8.618742300021034017279705745890, −7.29483257112871717385210954661, −7.04706058285827742325853352759, −5.99663473348686494527485846271, −5.04196054689156894877290791529, −4.35875205310566031724478881951, −2.68711232156862730021334875811, −1.67294995799890099500178310097,
0.817362822238807227889048621966, 2.20762196098433103329183260397, 3.41739203657913692311312132775, 3.73176105542191870815038758064, 5.55464873414498186061325447842, 6.17027944335911458680071024996, 7.20412905352214983559746816137, 7.938009727512618033438603945605, 8.938506493638308235428682768822, 10.00914089275950369346408953559
