L(s) = 1 | + 3i·3-s − i·7-s − 9·9-s − 42·11-s − 67i·13-s − 54i·17-s − 115·19-s + 3·21-s + 162i·23-s − 27i·27-s + 210·29-s + 193·31-s − 126i·33-s + 286i·37-s + 201·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.0539i·7-s − 0.333·9-s − 1.15·11-s − 1.42i·13-s − 0.770i·17-s − 1.38·19-s + 0.0311·21-s + 1.46i·23-s − 0.192i·27-s + 1.34·29-s + 1.11·31-s − 0.664i·33-s + 1.27i·37-s + 0.825·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.442047289\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.442047289\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + iT - 343T^{2} \) |
| 11 | \( 1 + 42T + 1.33e3T^{2} \) |
| 13 | \( 1 + 67iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 54iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 115T + 6.85e3T^{2} \) |
| 23 | \( 1 - 162iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 210T + 2.43e4T^{2} \) |
| 31 | \( 1 - 193T + 2.97e4T^{2} \) |
| 37 | \( 1 - 286iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 12T + 6.89e4T^{2} \) |
| 43 | \( 1 + 263iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 414iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 192iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 690T + 2.05e5T^{2} \) |
| 61 | \( 1 + 733T + 2.26e5T^{2} \) |
| 67 | \( 1 - 299iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 228T + 3.57e5T^{2} \) |
| 73 | \( 1 - 938iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 160T + 4.93e5T^{2} \) |
| 83 | \( 1 - 462iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 240T + 7.04e5T^{2} \) |
| 97 | \( 1 - 511iT - 9.12e5T^{2} \) |
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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
−9.761836937519103171611012461788, −8.541474982401347227654014230095, −8.078744092627659192562006052269, −7.12148276795583232838510876792, −5.97952787468616285686080369476, −5.22462872984904700318983263336, −4.45896263066938463554294840083, −3.21226214650368216767803415266, −2.51198740043673519225312159829, −0.77327502140195424191626100072,
0.46448602788039276588287141850, 1.95611203160081603962564116070, 2.64437790840564291877786052782, 4.12690351616618969844920618144, 4.86019748160008048949820800743, 6.18221641425964079490542097212, 6.57555185584490663999418252963, 7.62798733774925553933358993244, 8.463137815332931926072856704930, 8.945668163689817061408387429813