| L(s) = 1 | − 2.52·2-s + 4.38·4-s + 3.70·7-s − 6.03·8-s + 5.99·11-s − 0.118·13-s − 9.37·14-s + 6.47·16-s − 4.40·17-s + 4.28·19-s − 15.1·22-s + 6.92·23-s + 0.299·26-s + 16.2·28-s + 0.451·29-s + 9.98·31-s − 4.29·32-s + 11.1·34-s − 37-s − 10.8·38-s − 4.01·41-s − 3.58·43-s + 26.3·44-s − 17.5·46-s + 10.1·47-s + 6.74·49-s − 0.519·52-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 2.19·4-s + 1.40·7-s − 2.13·8-s + 1.80·11-s − 0.0328·13-s − 2.50·14-s + 1.61·16-s − 1.06·17-s + 0.981·19-s − 3.23·22-s + 1.44·23-s + 0.0587·26-s + 3.07·28-s + 0.0837·29-s + 1.79·31-s − 0.758·32-s + 1.90·34-s − 0.164·37-s − 1.75·38-s − 0.626·41-s − 0.546·43-s + 3.96·44-s − 2.58·46-s + 1.47·47-s + 0.964·49-s − 0.0720·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.396557658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.396557658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 7 | \( 1 - 3.70T + 7T^{2} \) |
| 11 | \( 1 - 5.99T + 11T^{2} \) |
| 13 | \( 1 + 0.118T + 13T^{2} \) |
| 17 | \( 1 + 4.40T + 17T^{2} \) |
| 19 | \( 1 - 4.28T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 0.451T + 29T^{2} \) |
| 31 | \( 1 - 9.98T + 31T^{2} \) |
| 41 | \( 1 + 4.01T + 41T^{2} \) |
| 43 | \( 1 + 3.58T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 + 2.69T + 59T^{2} \) |
| 61 | \( 1 + 4.97T + 61T^{2} \) |
| 67 | \( 1 + 6.11T + 67T^{2} \) |
| 71 | \( 1 - 15.9T + 71T^{2} \) |
| 73 | \( 1 - 8.89T + 73T^{2} \) |
| 79 | \( 1 + 2.69T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 3.95T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−7.991778890797784480720944473367, −7.22502554953686578088675917235, −6.78330179136766948188598930292, −6.12091021218831261615479209307, −4.98171125235799759734449736690, −4.34665239851530116484367009705, −3.19131602611021338612911670406, −2.19747777512753274340740710642, −1.38003346760484396361259141954, −0.889116657318850639371596371071,
0.889116657318850639371596371071, 1.38003346760484396361259141954, 2.19747777512753274340740710642, 3.19131602611021338612911670406, 4.34665239851530116484367009705, 4.98171125235799759734449736690, 6.12091021218831261615479209307, 6.78330179136766948188598930292, 7.22502554953686578088675917235, 7.991778890797784480720944473367
