L(s) = 1 | + (−1 − 1.41i)3-s + (−1.00 + 2.82i)9-s − 4.89·11-s + 4.89·13-s + 3.46i·17-s + 3.46i·19-s + 6·23-s + (5.00 − 1.41i)27-s − 2.82i·29-s + 3.46i·31-s + (4.89 + 6.92i)33-s + 4.89·37-s + (−4.89 − 6.92i)39-s + 5.65i·41-s − 8.48i·43-s + ⋯ |
L(s) = 1 | + (−0.577 − 0.816i)3-s + (−0.333 + 0.942i)9-s − 1.47·11-s + 1.35·13-s + 0.840i·17-s + 0.794i·19-s + 1.25·23-s + (0.962 − 0.272i)27-s − 0.525i·29-s + 0.622i·31-s + (0.852 + 1.20i)33-s + 0.805·37-s + (−0.784 − 1.10i)39-s + 0.883i·41-s − 1.29i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.180404614\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.180404614\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8.48iT - 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 5.65iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.954455997655903019835393355253, −8.626546103178587390581912285562, −8.108217539165187156538115812936, −7.31896955369200122600312328160, −6.31285865637278628295550530124, −5.70583011448270234491638969920, −4.83301292442243581886206884061, −3.49414225528935483124573925259, −2.29069040869609445714004762243, −1.04553373541549380425500131675,
0.68888718498217436963196781908, 2.66869962165406348198435109199, 3.59102511912768982253783679702, 4.78466107278159807219195134980, 5.32013987318024694124460409520, 6.26481220095622331829477383714, 7.18172722357959018802558510453, 8.245565112427424774418190725260, 9.042168862519953828001876225483, 9.771441199095287681757574334361