L(s) = 1 | − 0.517i·5-s − 1.73·13-s − 1.93i·17-s + 0.732·25-s − 1.93i·29-s + 37-s − 1.41i·41-s − 49-s + 1.41i·53-s − 61-s + 0.896i·65-s + 1.73·73-s − 0.999·85-s − 0.517i·89-s − 1.41i·101-s + ⋯ |
L(s) = 1 | − 0.517i·5-s − 1.73·13-s − 1.93i·17-s + 0.732·25-s − 1.93i·29-s + 37-s − 1.41i·41-s − 49-s + 1.41i·53-s − 61-s + 0.896i·65-s + 1.73·73-s − 0.999·85-s − 0.517i·89-s − 1.41i·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9475284197\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9475284197\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 0.517iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.73T + T^{2} \) |
| 17 | \( 1 + 1.93iT - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.93iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.73T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 0.517iT - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.115608813888797788733133216575, −7.955390598617305032789371002168, −7.45305543192013810005183867126, −6.69828508078383832763299308749, −5.62237848383146035782719088784, −4.86968635265284624672809973677, −4.35198226886831288760133793622, −2.93985134881242796767200467686, −2.25831577822435253588703897553, −0.60899951573795719134819383799,
1.60426420859416937580619331853, 2.68312597699913301943820190671, 3.52083208282787091214545049532, 4.59757079252904059187582916061, 5.28532270035516728105682527375, 6.36966700705006838694014124425, 6.87591225350693880975155221216, 7.77824226955588669886875152640, 8.374568084331339929906664857617, 9.360369926462881852974788974046