| L(s) = 1 | − 3-s + 1.50·7-s + 9-s − 6.17·11-s − 3.55·13-s + 0.495·17-s + 0.311·19-s − 1.50·21-s − 3.06·23-s − 27-s − 0.122·29-s − 2.94·31-s + 6.17·33-s − 4.36·37-s + 3.55·39-s + 4.25·41-s − 3.62·43-s + 5.28·47-s − 4.73·49-s − 0.495·51-s − 8.59·53-s − 0.311·57-s + 12.8·59-s − 11.3·61-s + 1.50·63-s + 13.2·67-s + 3.06·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.568·7-s + 0.333·9-s − 1.86·11-s − 0.986·13-s + 0.120·17-s + 0.0715·19-s − 0.328·21-s − 0.638·23-s − 0.192·27-s − 0.0227·29-s − 0.528·31-s + 1.07·33-s − 0.717·37-s + 0.569·39-s + 0.664·41-s − 0.553·43-s + 0.770·47-s − 0.676·49-s − 0.0694·51-s − 1.18·53-s − 0.0412·57-s + 1.67·59-s − 1.44·61-s + 0.189·63-s + 1.61·67-s + 0.368·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9260456600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9260456600\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.50T + 7T^{2} \) |
| 11 | \( 1 + 6.17T + 11T^{2} \) |
| 13 | \( 1 + 3.55T + 13T^{2} \) |
| 17 | \( 1 - 0.495T + 17T^{2} \) |
| 19 | \( 1 - 0.311T + 19T^{2} \) |
| 23 | \( 1 + 3.06T + 23T^{2} \) |
| 29 | \( 1 + 0.122T + 29T^{2} \) |
| 31 | \( 1 + 2.94T + 31T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 - 4.25T + 41T^{2} \) |
| 43 | \( 1 + 3.62T + 43T^{2} \) |
| 47 | \( 1 - 5.28T + 47T^{2} \) |
| 53 | \( 1 + 8.59T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 3.26T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 + 8.52T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.85965201706563480285257179542, −7.30853441299138102084873952515, −6.50217290791468854057683738366, −5.55217922328028775077552243645, −5.15791331832895134974366606633, −4.59939317483942409688311028517, −3.55491224307786027371186127664, −2.54825323569614599464094428253, −1.88458536207380353140636200217, −0.47564896801073206871782439352,
0.47564896801073206871782439352, 1.88458536207380353140636200217, 2.54825323569614599464094428253, 3.55491224307786027371186127664, 4.59939317483942409688311028517, 5.15791331832895134974366606633, 5.55217922328028775077552243645, 6.50217290791468854057683738366, 7.30853441299138102084873952515, 7.85965201706563480285257179542
