| L(s) = 1 | − 2.79·2-s + 3-s + 5.82·4-s − 2.79·6-s − 3.22·7-s − 10.7·8-s + 9-s − 1.78·11-s + 5.82·12-s − 2.06·13-s + 9.02·14-s + 18.2·16-s + 4.80·17-s − 2.79·18-s − 4.95·19-s − 3.22·21-s + 4.98·22-s + 6.09·23-s − 10.7·24-s + 5.78·26-s + 27-s − 18.7·28-s − 5.59·29-s − 1.05·31-s − 29.7·32-s − 1.78·33-s − 13.4·34-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 0.577·3-s + 2.91·4-s − 1.14·6-s − 1.21·7-s − 3.78·8-s + 0.333·9-s − 0.537·11-s + 1.68·12-s − 0.573·13-s + 2.41·14-s + 4.57·16-s + 1.16·17-s − 0.659·18-s − 1.13·19-s − 0.703·21-s + 1.06·22-s + 1.27·23-s − 2.18·24-s + 1.13·26-s + 0.192·27-s − 3.55·28-s − 1.03·29-s − 0.189·31-s − 5.26·32-s − 0.310·33-s − 2.30·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5709142092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5709142092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 107 | \( 1 + T \) |
| good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 7 | \( 1 + 3.22T + 7T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 13 | \( 1 + 2.06T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 + 4.95T + 19T^{2} \) |
| 23 | \( 1 - 6.09T + 23T^{2} \) |
| 29 | \( 1 + 5.59T + 29T^{2} \) |
| 31 | \( 1 + 1.05T + 31T^{2} \) |
| 37 | \( 1 - 1.10T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 + 4.41T + 47T^{2} \) |
| 53 | \( 1 + 1.51T + 53T^{2} \) |
| 59 | \( 1 - 5.64T + 59T^{2} \) |
| 61 | \( 1 + 6.43T + 61T^{2} \) |
| 67 | \( 1 + 6.76T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 1.06T + 73T^{2} \) |
| 79 | \( 1 + 5.83T + 79T^{2} \) |
| 83 | \( 1 + 7.59T + 83T^{2} \) |
| 89 | \( 1 + 5.58T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.86985548340012254501044516248, −7.40527912331425394400480692065, −6.82273404639947239481189314024, −6.12419314738766717513743817261, −5.40245391797760500835886134449, −3.89103705971290461022862666695, −2.92976078025842039834509135893, −2.63208289752094387418884367499, −1.57121335286914823023177398651, −0.48334827290845128903590531849,
0.48334827290845128903590531849, 1.57121335286914823023177398651, 2.63208289752094387418884367499, 2.92976078025842039834509135893, 3.89103705971290461022862666695, 5.40245391797760500835886134449, 6.12419314738766717513743817261, 6.82273404639947239481189314024, 7.40527912331425394400480692065, 7.86985548340012254501044516248
