L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.5 − 0.363i)3-s + (−0.809 + 0.587i)4-s + (−0.5 − 0.363i)6-s + 1.61·7-s + (0.809 + 0.587i)8-s + (−0.190 + 0.587i)9-s + (−0.190 + 0.587i)12-s + (−0.500 − 1.53i)14-s + (0.309 − 0.951i)16-s + 0.618·18-s + (0.809 − 0.587i)21-s + (0.5 + 1.53i)23-s + 0.618·24-s + (0.309 + 0.951i)27-s + (−1.30 + 0.951i)28-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.5 − 0.363i)3-s + (−0.809 + 0.587i)4-s + (−0.5 − 0.363i)6-s + 1.61·7-s + (0.809 + 0.587i)8-s + (−0.190 + 0.587i)9-s + (−0.190 + 0.587i)12-s + (−0.500 − 1.53i)14-s + (0.309 − 0.951i)16-s + 0.618·18-s + (0.809 − 0.587i)21-s + (0.5 + 1.53i)23-s + 0.618·24-s + (0.309 + 0.951i)27-s + (−1.30 + 0.951i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.340368738\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.340368738\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 - 1.61T + T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + 0.618T + T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
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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
−8.844204968869000733916277442958, −8.382496373593077911209054602450, −7.66379896253802393615747999157, −7.21216969146651158184987892094, −5.47647887680163473249673558240, −5.01359347097360965261668888613, −4.03920749211525859179218939814, −3.04189233844060612608802731490, −2.00222429803606701063238480600, −1.44181257772867618342750506408,
1.13390886438567413976883468019, 2.47492414731401688452165838901, 3.88636518146787945192765184401, 4.56780752508732923047161270625, 5.26913128329759864484970343098, 6.19810842539751469935907022040, 6.99858959943464981534159253512, 7.924238446708940344999826260354, 8.365122820761718523728367306073, 8.971388268367115401925969065488