| L(s) = 1 | − 2-s + 3-s + 4-s + 1.14·5-s − 6-s − 8-s + 9-s − 1.14·10-s − 2.16·11-s + 12-s − 2.29·13-s + 1.14·15-s + 16-s + 4.99·17-s − 18-s − 19-s + 1.14·20-s + 2.16·22-s − 6.91·23-s − 24-s − 3.67·25-s + 2.29·26-s + 27-s + 3.76·29-s − 1.14·30-s − 4.38·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.513·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.363·10-s − 0.653·11-s + 0.288·12-s − 0.637·13-s + 0.296·15-s + 0.250·16-s + 1.21·17-s − 0.235·18-s − 0.229·19-s + 0.256·20-s + 0.462·22-s − 1.44·23-s − 0.204·24-s − 0.735·25-s + 0.450·26-s + 0.192·27-s + 0.699·29-s − 0.209·30-s − 0.787·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.14T + 5T^{2} \) |
| 11 | \( 1 + 2.16T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 17 | \( 1 - 4.99T + 17T^{2} \) |
| 23 | \( 1 + 6.91T + 23T^{2} \) |
| 29 | \( 1 - 3.76T + 29T^{2} \) |
| 31 | \( 1 + 4.38T + 31T^{2} \) |
| 37 | \( 1 - 0.660T + 37T^{2} \) |
| 41 | \( 1 + 0.806T + 41T^{2} \) |
| 43 | \( 1 + 2.08T + 43T^{2} \) |
| 47 | \( 1 - 3.72T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 2.57T + 61T^{2} \) |
| 67 | \( 1 - 8.88T + 67T^{2} \) |
| 71 | \( 1 + 0.323T + 71T^{2} \) |
| 73 | \( 1 - 4.72T + 73T^{2} \) |
| 79 | \( 1 + 7.59T + 79T^{2} \) |
| 83 | \( 1 + 3.17T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 5.91T + 97T^{2} \) |
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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
−7.84187684845282696813080846361, −7.41156216525170635740027543201, −6.40719679974778428091616346803, −5.76106331876439787359830040193, −4.97160487931646326432183391539, −3.92146802954181133914315874624, −3.01811589386946244167344807599, −2.24591202584379325525133125615, −1.47207023580028135728887638537, 0,
1.47207023580028135728887638537, 2.24591202584379325525133125615, 3.01811589386946244167344807599, 3.92146802954181133914315874624, 4.97160487931646326432183391539, 5.76106331876439787359830040193, 6.40719679974778428091616346803, 7.41156216525170635740027543201, 7.84187684845282696813080846361
