L(s) = 1 | + (0.654 + 0.755i)5-s + (0.888 + 0.458i)7-s + (−0.327 + 0.945i)11-s + (−0.786 − 0.618i)13-s + (0.995 + 0.0950i)17-s + (0.888 − 0.458i)19-s + (0.235 − 0.971i)23-s + (−0.142 + 0.989i)25-s + (0.5 + 0.866i)29-s + (0.786 − 0.618i)31-s + (0.235 + 0.971i)35-s + (−0.5 + 0.866i)37-s + (−0.580 + 0.814i)41-s + (−0.415 − 0.909i)43-s + (0.723 + 0.690i)47-s + ⋯ |
L(s) = 1 | + (0.654 + 0.755i)5-s + (0.888 + 0.458i)7-s + (−0.327 + 0.945i)11-s + (−0.786 − 0.618i)13-s + (0.995 + 0.0950i)17-s + (0.888 − 0.458i)19-s + (0.235 − 0.971i)23-s + (−0.142 + 0.989i)25-s + (0.5 + 0.866i)29-s + (0.786 − 0.618i)31-s + (0.235 + 0.971i)35-s + (−0.5 + 0.866i)37-s + (−0.580 + 0.814i)41-s + (−0.415 − 0.909i)43-s + (0.723 + 0.690i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.599055714 + 0.8642305451i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.599055714 + 0.8642305451i\) |
\(L(1)\) |
\(\approx\) |
\(1.274420262 + 0.3148485197i\) |
\(L(1)\) |
\(\approx\) |
\(1.274420262 + 0.3148485197i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 67 | \( 1 \) |
good | 5 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (0.888 + 0.458i)T \) |
| 11 | \( 1 + (-0.327 + 0.945i)T \) |
| 13 | \( 1 + (-0.786 - 0.618i)T \) |
| 17 | \( 1 + (0.995 + 0.0950i)T \) |
| 19 | \( 1 + (0.888 - 0.458i)T \) |
| 23 | \( 1 + (0.235 - 0.971i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.786 - 0.618i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.580 + 0.814i)T \) |
| 43 | \( 1 + (-0.415 - 0.909i)T \) |
| 47 | \( 1 + (0.723 + 0.690i)T \) |
| 53 | \( 1 + (-0.415 + 0.909i)T \) |
| 59 | \( 1 + (-0.142 - 0.989i)T \) |
| 61 | \( 1 + (-0.327 - 0.945i)T \) |
| 71 | \( 1 + (-0.995 + 0.0950i)T \) |
| 73 | \( 1 + (-0.327 - 0.945i)T \) |
| 79 | \( 1 + (-0.928 + 0.371i)T \) |
| 83 | \( 1 + (0.981 - 0.189i)T \) |
| 89 | \( 1 + (0.959 + 0.281i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.84038846709773746767110684604, −21.190404698528967374777579944793, −20.81870363772447430725806333257, −19.746841068772317293674497457577, −18.97428889746139340183755878405, −17.9459443759641264025362517948, −17.27160792135138473975737011797, −16.58775331208463090085835042980, −15.86669090499464880253238990108, −14.59215180754815019235850433158, −13.90045685134136549430498094089, −13.423762014450897941556635330957, −12.11798858880798068734547849749, −11.65879394652889366616613125473, −10.43129123480239004358426207058, −9.767319200054687671733493824388, −8.77435285122686882912582756283, −7.97278407379814156254610274397, −7.14201850768665135974595879671, −5.75945074434432478238768138482, −5.23246825339437763449892018403, −4.28439093029489026303399557445, −3.07684378868285831383150053555, −1.804447440869198664846903074049, −0.93700196300246786018442032524,
1.36890528282402939901673544255, 2.44215787698556100879884545261, 3.12151659543091379184348272734, 4.77587371919818459098542722084, 5.25259003633461625793292316087, 6.37876695067381410038760162593, 7.36486597375756813026795397545, 8.02642312233064985459809273434, 9.24457116262811225006453872425, 10.10240896280308380281895257085, 10.66182873879171448593289129548, 11.82678288822347690601597754497, 12.43915437555377427502335362071, 13.54847086410486997007606065260, 14.44075505790083445134973774187, 14.92906661856005084447700450045, 15.69905774473658601421181189840, 17.09026712805971816482748197246, 17.547939850079194267186320414323, 18.38822985493122203633692588362, 18.881544490773799965234594916087, 20.251526811053669183978085942, 20.698309329899646722040476614888, 21.73684250900019680302604602105, 22.2302277111169403382538890153