| L(s) = 1 | − 2.97·3-s + (2.14 + 0.615i)5-s + 2.64·7-s + 5.82·9-s + 7.17i·13-s + (−6.38 − 1.82i)15-s + 6.81i·19-s − 7.86·21-s − 7.48·23-s + (4.24 + 2.64i)25-s − 8.40·27-s + (5.68 + 1.62i)35-s − 21.3i·39-s + (12.5 + 3.58i)45-s + 7.00·49-s + ⋯ |
| L(s) = 1 | − 1.71·3-s + (0.961 + 0.275i)5-s + 0.999·7-s + 1.94·9-s + 1.98i·13-s + (−1.64 − 0.472i)15-s + 1.56i·19-s − 1.71·21-s − 1.56·23-s + (0.848 + 0.529i)25-s − 1.61·27-s + (0.961 + 0.275i)35-s − 3.41i·39-s + (1.86 + 0.534i)45-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9922559457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9922559457\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.14 - 0.615i)T \) |
| 7 | \( 1 - 2.64T \) |
| good | 3 | \( 1 + 2.97T + 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 7.17iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 6.81iT - 19T^{2} \) |
| 23 | \( 1 + 7.48T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 13.9iT - 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 16.9iT - 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.650648787944748983778371082340, −8.527930742401608512146548864230, −7.52556913831278246264323129895, −6.64862297009678540785244679233, −6.11411827247510626052375809643, −5.49364585079609956025881874754, −4.64844358313027009298214162793, −3.96726278630930078989703238821, −1.97153314519423423621693039257, −1.49081555825325346608014738486,
0.45026801616159895613976808015, 1.44344522957845412671821676380, 2.70269501270809372049969605021, 4.36009643313592500734254433598, 5.02026259969752100049529425387, 5.65558822060470855452112579307, 6.06624203764648279791757333959, 7.10452213303476248536284050394, 7.895616392373532719170874865373, 8.805011233152949088522725801752
