| L(s) = 1 | − 0.183·2-s − 1.47·3-s − 1.96·4-s + 0.269·6-s − 3.26·7-s + 0.726·8-s − 0.833·9-s + 2·11-s + 2.89·12-s − 0.296·13-s + 0.597·14-s + 3.79·16-s − 5.16·17-s + 0.152·18-s + 1.73·19-s + 4.79·21-s − 0.366·22-s + 0.879·23-s − 1.06·24-s + 0.0542·26-s + 5.64·27-s + 6.41·28-s + 5.91·29-s + 6.09·31-s − 2.14·32-s − 2.94·33-s + 0.945·34-s + ⋯ |
| L(s) = 1 | − 0.129·2-s − 0.849·3-s − 0.983·4-s + 0.110·6-s − 1.23·7-s + 0.256·8-s − 0.277·9-s + 0.603·11-s + 0.835·12-s − 0.0822·13-s + 0.159·14-s + 0.949·16-s − 1.25·17-s + 0.0359·18-s + 0.396·19-s + 1.04·21-s − 0.0781·22-s + 0.183·23-s − 0.218·24-s + 0.0106·26-s + 1.08·27-s + 1.21·28-s + 1.09·29-s + 1.09·31-s − 0.379·32-s − 0.512·33-s + 0.162·34-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
| \(L(1)\) |
\(\approx\) |
\(0.5403898818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5403898818\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.183T + 2T^{2} \) |
| 3 | \( 1 + 1.47T + 3T^{2} \) |
| 7 | \( 1 + 3.26T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 0.296T + 13T^{2} \) |
| 17 | \( 1 + 5.16T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 - 0.879T + 23T^{2} \) |
| 29 | \( 1 - 5.91T + 29T^{2} \) |
| 31 | \( 1 - 6.09T + 31T^{2} \) |
| 37 | \( 1 - 8.08T + 37T^{2} \) |
| 41 | \( 1 + 1.01T + 41T^{2} \) |
| 43 | \( 1 + 3.24T + 43T^{2} \) |
| 47 | \( 1 - 4.21T + 47T^{2} \) |
| 53 | \( 1 + 8.10T + 53T^{2} \) |
| 59 | \( 1 - 5.93T + 59T^{2} \) |
| 61 | \( 1 - 0.915T + 61T^{2} \) |
| 67 | \( 1 - 6.88T + 67T^{2} \) |
| 71 | \( 1 + 5.96T + 71T^{2} \) |
| 73 | \( 1 - 8.83T + 73T^{2} \) |
| 79 | \( 1 + 7.76T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 7.52T + 89T^{2} \) |
| 97 | \( 1 + 6.72T + 97T^{2} \) |
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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
−10.50831360727110719411590287129, −9.695788875898770019388281794447, −9.037910095680878686824937678504, −8.152916015404425008544251299158, −6.70928293049098900350801578324, −6.19794896497394275594154582474, −5.07706112551006603624199562607, −4.17550371593200739780692679367, −2.93567704606131208737572349084, −0.66639282546174732916322857934,
0.66639282546174732916322857934, 2.93567704606131208737572349084, 4.17550371593200739780692679367, 5.07706112551006603624199562607, 6.19794896497394275594154582474, 6.70928293049098900350801578324, 8.152916015404425008544251299158, 9.037910095680878686824937678504, 9.695788875898770019388281794447, 10.50831360727110719411590287129
