L(s) = 1 | − 0.500·2-s − 3.09·3-s − 1.74·4-s + 1.54·6-s − 0.0237·7-s + 1.87·8-s + 6.56·9-s + 3.58·11-s + 5.41·12-s − 3.77·13-s + 0.0119·14-s + 2.55·16-s + 3.62·17-s − 3.29·18-s − 2.43·19-s + 0.0735·21-s − 1.79·22-s − 1.71·23-s − 5.80·24-s + 1.89·26-s − 11.0·27-s + 0.0416·28-s + 3.85·29-s − 6.00·31-s − 5.03·32-s − 11.0·33-s − 1.81·34-s + ⋯ |
L(s) = 1 | − 0.354·2-s − 1.78·3-s − 0.874·4-s + 0.632·6-s − 0.00899·7-s + 0.664·8-s + 2.18·9-s + 1.08·11-s + 1.56·12-s − 1.04·13-s + 0.00318·14-s + 0.639·16-s + 0.878·17-s − 0.775·18-s − 0.557·19-s + 0.0160·21-s − 0.382·22-s − 0.357·23-s − 1.18·24-s + 0.370·26-s − 2.12·27-s + 0.00786·28-s + 0.716·29-s − 1.07·31-s − 0.890·32-s − 1.92·33-s − 0.311·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + 0.500T + 2T^{2} \) |
| 3 | \( 1 + 3.09T + 3T^{2} \) |
| 7 | \( 1 + 0.0237T + 7T^{2} \) |
| 11 | \( 1 - 3.58T + 11T^{2} \) |
| 13 | \( 1 + 3.77T + 13T^{2} \) |
| 17 | \( 1 - 3.62T + 17T^{2} \) |
| 19 | \( 1 + 2.43T + 19T^{2} \) |
| 23 | \( 1 + 1.71T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 + 6.00T + 31T^{2} \) |
| 37 | \( 1 - 0.369T + 37T^{2} \) |
| 41 | \( 1 + 7.80T + 41T^{2} \) |
| 43 | \( 1 + 0.174T + 43T^{2} \) |
| 47 | \( 1 + 7.81T + 47T^{2} \) |
| 53 | \( 1 - 8.97T + 53T^{2} \) |
| 59 | \( 1 + 4.45T + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 - 4.47T + 67T^{2} \) |
| 71 | \( 1 + 9.69T + 71T^{2} \) |
| 73 | \( 1 - 3.95T + 73T^{2} \) |
| 79 | \( 1 + 9.68T + 79T^{2} \) |
| 83 | \( 1 - 8.95T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 + 2.76T + 97T^{2} \) |
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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
−10.04827067349118552401677551729, −9.702542145687778234285829379033, −8.487939355859804371335973002902, −7.33533075669017918195299589320, −6.51261091193478804594576961859, −5.48687985356473358011645027700, −4.79274851814194741141831435953, −3.85789873327660862018745682965, −1.39240663451572315699153817465, 0,
1.39240663451572315699153817465, 3.85789873327660862018745682965, 4.79274851814194741141831435953, 5.48687985356473358011645027700, 6.51261091193478804594576961859, 7.33533075669017918195299589320, 8.487939355859804371335973002902, 9.702542145687778234285829379033, 10.04827067349118552401677551729