L(s) = 1 | + (−0.309 − 0.951i)3-s + (0.809 + 0.587i)5-s + (0.309 − 0.951i)7-s + (−0.809 + 0.587i)9-s + (−0.809 + 0.587i)13-s + (0.309 − 0.951i)15-s + (0.809 + 0.587i)17-s + (0.309 + 0.951i)19-s − 21-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (0.309 − 0.951i)29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)35-s + (−0.309 + 0.951i)37-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)3-s + (0.809 + 0.587i)5-s + (0.309 − 0.951i)7-s + (−0.809 + 0.587i)9-s + (−0.809 + 0.587i)13-s + (0.309 − 0.951i)15-s + (0.809 + 0.587i)17-s + (0.309 + 0.951i)19-s − 21-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (0.309 − 0.951i)29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)35-s + (−0.309 + 0.951i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.547905865 - 0.2816068452i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.547905865 - 0.2816068452i\) |
\(L(1)\) |
\(\approx\) |
\(1.125985361 - 0.2084410261i\) |
\(L(1)\) |
\(\approx\) |
\(1.125985361 - 0.2084410261i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.809 + 0.587i)T \) |
| 7 | \( 1 + (0.309 - 0.951i)T \) |
| 13 | \( 1 + (-0.809 + 0.587i)T \) |
| 17 | \( 1 + (0.809 + 0.587i)T \) |
| 19 | \( 1 + (0.309 + 0.951i)T \) |
| 29 | \( 1 + (0.309 - 0.951i)T \) |
| 31 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.309 + 0.951i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.309 - 0.951i)T \) |
| 53 | \( 1 + (0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.309 + 0.951i)T \) |
| 61 | \( 1 + (0.809 + 0.587i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.809 + 0.587i)T \) |
| 73 | \( 1 + (0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.809 + 0.587i)T \) |
| 83 | \( 1 + (-0.809 - 0.587i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.809 - 0.587i)T \) |
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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
−21.61807867467349716567949917973, −21.072482957857512421657929920979, −20.33548997711102265837222242655, −19.499195548864843794129192433100, −18.29474089811996733902623964502, −17.567720453340383269244004993089, −17.11287024711346402664519766497, −15.966159872368217925179583046887, −15.65720952270197296516826503604, −14.48062098686377360307551276698, −14.07101521357030013552813423910, −12.64214422409416363011698886993, −12.22193069819276275124735745805, −11.20951710960730593607819585239, −10.322936543009509018980908758579, −9.48810939271251238464778090580, −9.034778309487139666344323508440, −8.09853441947867130543513672799, −6.79475428614424615549659553610, −5.54909692254242436696610603512, −5.31790422876224940106915766603, −4.503834144146568963977746198501, −3.07893704616334326704665924227, −2.36357514249975231606337522467, −0.86110160682589316214284622812,
1.04607785874390698389223541814, 1.90314391530550274998341042502, 2.84115222735003761589625047471, 4.08448934958080583324864173389, 5.2665773095184636991617623007, 6.107391344839681178667784453071, 6.856877349190126776417969399839, 7.600840751057214556921771745369, 8.33595852470234675988703563068, 9.838399653573745884008137582112, 10.212939181059501883771414086712, 11.30318741741615012823979074970, 11.97466108416147556177880141938, 12.95303173335678366703737808416, 13.74418356767625071014205977464, 14.24338125322603714042461820963, 14.951865720479388504635896657315, 16.52637537149167015384704878458, 17.044282800852564624297046984301, 17.58534171884257354203353642564, 18.49344830341154308086768575420, 19.09116859667257645165905067249, 19.85802037822316122112289052420, 20.90989892582296028294573545712, 21.51847320736123761611881269624