L(s) = 1 | + (0.5 + 0.866i)2-s + (0.104 + 0.994i)3-s + (−0.5 + 0.866i)4-s + (−0.809 + 0.587i)6-s + (−0.669 + 0.743i)7-s − 8-s + (−0.978 + 0.207i)9-s + (−0.5 − 0.866i)11-s + (−0.913 − 0.406i)12-s + (0.104 + 0.994i)13-s + (−0.978 − 0.207i)14-s + (−0.5 − 0.866i)16-s + (−0.309 − 0.951i)17-s + (−0.669 − 0.743i)18-s + (−0.5 − 0.866i)19-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (0.104 + 0.994i)3-s + (−0.5 + 0.866i)4-s + (−0.809 + 0.587i)6-s + (−0.669 + 0.743i)7-s − 8-s + (−0.978 + 0.207i)9-s + (−0.5 − 0.866i)11-s + (−0.913 − 0.406i)12-s + (0.104 + 0.994i)13-s + (−0.978 − 0.207i)14-s + (−0.5 − 0.866i)16-s + (−0.309 − 0.951i)17-s + (−0.669 − 0.743i)18-s + (−0.5 − 0.866i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1780268332 - 0.03583030841i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1780268332 - 0.03583030841i\) |
\(L(1)\) |
\(\approx\) |
\(0.5979633259 + 0.6463472844i\) |
\(L(1)\) |
\(\approx\) |
\(0.5979633259 + 0.6463472844i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.104 + 0.994i)T \) |
| 7 | \( 1 + (-0.669 + 0.743i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.104 + 0.994i)T \) |
| 17 | \( 1 + (-0.309 - 0.951i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.809 - 0.587i)T \) |
| 29 | \( 1 + (-0.978 + 0.207i)T \) |
| 31 | \( 1 + (0.669 - 0.743i)T \) |
| 37 | \( 1 + (0.104 - 0.994i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 + (-0.309 + 0.951i)T \) |
| 47 | \( 1 + (-0.309 + 0.951i)T \) |
| 53 | \( 1 + (-0.669 - 0.743i)T \) |
| 59 | \( 1 + (-0.104 + 0.994i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 + (-0.669 + 0.743i)T \) |
| 71 | \( 1 + (-0.978 + 0.207i)T \) |
| 73 | \( 1 + (0.809 + 0.587i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (0.104 - 0.994i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.669 - 0.743i)T \) |
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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
−20.93585142341057314052483135078, −20.38779600003867959487884032830, −19.73733552965155060674004937252, −19.09013692640332699024206192798, −18.41068037888469024132306668574, −17.49503821412365024948294475311, −16.98096418061086653789507957345, −15.42583514490791773013711661144, −14.956366063940648628898806890066, −13.85090868663610160263722740912, −13.29126737540910920691374035874, −12.63399467863099304993754048754, −12.2363874699886341173495701244, −10.9673697293425988490711291847, −10.43090040691055851993832365507, −9.594859829856692907233763203356, −8.50369608915671419334832828384, −7.62981817458950603619982275808, −6.68337531276167557055956557745, −5.90649000450110764605308170641, −4.98599438905732643173852736789, −3.77772825483566563450423866531, −3.11264440322274724139261182249, −2.05987968725289171174093336063, −1.23936615039858403502417749565,
0.060553820937023576491244686256, 2.635610839823834376749994097731, 3.04680817621600188817450855141, 4.24526185551680746021475534967, 4.858543600801568966451585019280, 5.81702595083537567042140147263, 6.41100352660711859524457031903, 7.48352190808304753514218763281, 8.67998936474384799199253832044, 9.020436661773811589021976090066, 9.77198453557758195704811887743, 11.16710583677901502137588256153, 11.58820711461962777761105509262, 12.87984825138201543344862821614, 13.43942755268013285536953530703, 14.381228972817318924763928613504, 15.026666409624265575844589515033, 15.822908095037169800709812283536, 16.29593704412658284555168731249, 16.83502689304926205529565946710, 17.92920759936955766933134119043, 18.77193973492073493621204039501, 19.54205598150841735965220839211, 20.821979952873775794634635534, 21.28095693673227375629814702503