L(s) = 1 | − i·2-s − 4-s + (0.707 − 0.707i)7-s + i·8-s + (1.70 + 0.707i)11-s + (−0.707 − 0.707i)14-s + 16-s + (0.707 − 1.70i)22-s + (−1.41 + 1.41i)23-s + (0.707 + 0.707i)25-s + (−0.707 + 0.707i)28-s + (0.292 + 0.707i)29-s − i·32-s + (0.292 − 0.707i)37-s + (0.707 − 1.70i)43-s + (−1.70 − 0.707i)44-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + (0.707 − 0.707i)7-s + i·8-s + (1.70 + 0.707i)11-s + (−0.707 − 0.707i)14-s + 16-s + (0.707 − 1.70i)22-s + (−1.41 + 1.41i)23-s + (0.707 + 0.707i)25-s + (−0.707 + 0.707i)28-s + (0.292 + 0.707i)29-s − i·32-s + (0.292 − 0.707i)37-s + (0.707 − 1.70i)43-s + (−1.70 − 0.707i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.231432303\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.231432303\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 5 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 29 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 + (1 + i)T + iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.272433852056971904454434217881, −8.706971656062340280677902619193, −7.66006900368811252627599110127, −7.02881289733592091068897327658, −5.83548954789336092999128284641, −4.87776530724333982973577493149, −4.03856363175110371416968648058, −3.54258405767835235101842307842, −1.98387736859633511714752432772, −1.27611551381802707517741574642,
1.22051328488457987405152735530, 2.77822772770271093566246498602, 4.17749426189355032103994345603, 4.53595382149434103466689047918, 5.88248638198901120203355381148, 6.17759469712873789512885132154, 7.02329640077634459901680003860, 8.216652308879171791732661144170, 8.430443641619151135139726918793, 9.230020116679288672061604353976