L(s) = 1 | − i·5-s + (0.585 − 2.58i)7-s + 3.02i·11-s − 0.829i·13-s − 0.756i·17-s − 2.22·19-s − 0.0727i·23-s − 25-s + 4.71·29-s + 0.264·31-s + (−2.58 − 0.585i)35-s − 4.13·37-s − 7.42i·41-s − 9.01i·43-s − 3.88·47-s + ⋯ |
L(s) = 1 | − 0.447i·5-s + (0.221 − 0.975i)7-s + 0.910i·11-s − 0.229i·13-s − 0.183i·17-s − 0.509·19-s − 0.0151i·23-s − 0.200·25-s + 0.876·29-s + 0.0475·31-s + (−0.436 − 0.0989i)35-s − 0.680·37-s − 1.15i·41-s − 1.37i·43-s − 0.567·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.237125370\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.237125370\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-0.585 + 2.58i)T \) |
good | 11 | \( 1 - 3.02iT - 11T^{2} \) |
| 13 | \( 1 + 0.829iT - 13T^{2} \) |
| 17 | \( 1 + 0.756iT - 17T^{2} \) |
| 19 | \( 1 + 2.22T + 19T^{2} \) |
| 23 | \( 1 + 0.0727iT - 23T^{2} \) |
| 29 | \( 1 - 4.71T + 29T^{2} \) |
| 31 | \( 1 - 0.264T + 31T^{2} \) |
| 37 | \( 1 + 4.13T + 37T^{2} \) |
| 41 | \( 1 + 7.42iT - 41T^{2} \) |
| 43 | \( 1 + 9.01iT - 43T^{2} \) |
| 47 | \( 1 + 3.88T + 47T^{2} \) |
| 53 | \( 1 - 4.40T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 0.745iT - 61T^{2} \) |
| 67 | \( 1 + 4.48iT - 67T^{2} \) |
| 71 | \( 1 - 12.0iT - 71T^{2} \) |
| 73 | \( 1 + 7.24iT - 73T^{2} \) |
| 79 | \( 1 + 2.57iT - 79T^{2} \) |
| 83 | \( 1 + 4.52T + 83T^{2} \) |
| 89 | \( 1 + 9.82iT - 89T^{2} \) |
| 97 | \( 1 + 17.5iT - 97T^{2} \) |
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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
−8.013562413344282832535035077648, −7.09855574305040221889946161059, −6.84083783106593473073711987885, −5.67704347834159787716317743708, −4.95126332862157028518943498248, −4.27421186994400670638578691187, −3.60822886757755361217756294332, −2.39763140997362316452896124935, −1.47166609913581779876132949070, −0.33718222555137846325959125048,
1.27943825696674243719660130528, 2.41614629838143942806403119829, 3.04131883823730423343034075572, 3.98854124159595961067988483840, 4.92314831205583935648554338942, 5.66708874676000687698537765808, 6.36136127643861203157150673949, 6.87828008633229752048782426737, 8.128200456921143688339593955968, 8.278230723148997285330700415092