L(s) = 1 | + (−0.982 − 0.183i)2-s + (−0.602 + 0.798i)3-s + (0.932 + 0.361i)4-s + (−0.982 + 0.183i)5-s + (0.739 − 0.673i)6-s + (0.445 − 0.895i)7-s + (−0.850 − 0.526i)8-s + (−0.273 − 0.961i)9-s + 10-s + (0.932 + 0.361i)11-s + (−0.850 + 0.526i)12-s + (0.445 − 0.895i)13-s + (−0.602 + 0.798i)14-s + (0.445 − 0.895i)15-s + (0.739 + 0.673i)16-s + (−0.850 + 0.526i)17-s + ⋯ |
L(s) = 1 | + (−0.982 − 0.183i)2-s + (−0.602 + 0.798i)3-s + (0.932 + 0.361i)4-s + (−0.982 + 0.183i)5-s + (0.739 − 0.673i)6-s + (0.445 − 0.895i)7-s + (−0.850 − 0.526i)8-s + (−0.273 − 0.961i)9-s + 10-s + (0.932 + 0.361i)11-s + (−0.850 + 0.526i)12-s + (0.445 − 0.895i)13-s + (−0.602 + 0.798i)14-s + (0.445 − 0.895i)15-s + (0.739 + 0.673i)16-s + (−0.850 + 0.526i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5180799753 + 0.09511034930i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5180799753 + 0.09511034930i\) |
\(L(1)\) |
\(\approx\) |
\(0.5626573656 + 0.06600797267i\) |
\(L(1)\) |
\(\approx\) |
\(0.5626573656 + 0.06600797267i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 137 | \( 1 \) |
good | 2 | \( 1 + (-0.982 - 0.183i)T \) |
| 3 | \( 1 + (-0.602 + 0.798i)T \) |
| 5 | \( 1 + (-0.982 + 0.183i)T \) |
| 7 | \( 1 + (0.445 - 0.895i)T \) |
| 11 | \( 1 + (0.932 + 0.361i)T \) |
| 13 | \( 1 + (0.445 - 0.895i)T \) |
| 17 | \( 1 + (-0.850 + 0.526i)T \) |
| 19 | \( 1 + (0.0922 + 0.995i)T \) |
| 23 | \( 1 + (0.739 + 0.673i)T \) |
| 29 | \( 1 + (0.739 + 0.673i)T \) |
| 31 | \( 1 + (0.0922 - 0.995i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (0.0922 + 0.995i)T \) |
| 47 | \( 1 + (-0.273 - 0.961i)T \) |
| 53 | \( 1 + (0.0922 + 0.995i)T \) |
| 59 | \( 1 + (-0.273 - 0.961i)T \) |
| 61 | \( 1 + (-0.273 + 0.961i)T \) |
| 67 | \( 1 + (0.445 - 0.895i)T \) |
| 71 | \( 1 + (0.932 - 0.361i)T \) |
| 73 | \( 1 + (0.445 + 0.895i)T \) |
| 79 | \( 1 + (-0.602 - 0.798i)T \) |
| 83 | \( 1 + (-0.850 - 0.526i)T \) |
| 89 | \( 1 + (-0.982 + 0.183i)T \) |
| 97 | \( 1 + (0.932 + 0.361i)T \) |
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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
−28.44382770752612774785916473415, −27.52288414528422979113617772138, −26.725466669588539540934873828756, −25.2559594488877667958343003598, −24.45394305059873561449297514773, −23.8974423614791016146896000947, −22.66626341255088303333940473828, −21.358766564422425555092417970676, −19.87363747608073650166094235067, −19.17993578480122666901949129498, −18.37532247502379319970552559595, −17.420626110069326857479051341898, −16.356107773756823855588786161391, −15.53197989005341482667364425099, −14.20598012915251512659530433482, −12.47777121262610301024881521551, −11.42080183125158949015855465917, −11.21712354550763707856108530271, −9.05083305189969189763581849153, −8.41805290067844041789749037983, −7.10184625876155439988218211428, −6.3103321426912488603719369674, −4.75793774299311384411349896414, −2.51807686749374243619083849884, −1.00016630660112949470569367669,
1.034556922895572079534200483264, 3.41532225425826693835256349231, 4.33684412437557111371226642798, 6.23739765864995738417213251765, 7.428642122329920929024203764282, 8.5238780775395621564357687739, 9.82764360152777515636981369525, 10.883844979931222124795259269873, 11.39241933357126896925846113139, 12.58612616859054108099017668914, 14.70167833697460426270821849717, 15.49543975348831856226425610383, 16.55041650426194742057291486934, 17.31659992368103061763501417926, 18.25183120111865579431325381963, 19.76667472524705114255805231661, 20.22273841285322560641506235138, 21.343081936090258013753613220418, 22.64676382177839490392121903425, 23.44344935487708590885982899000, 24.68002182694883561147591677664, 26.01657302019772617896406479740, 26.976939446696112899583447587701, 27.44322373731486378812473222371, 28.12721824684675948937479345229