| L(s) = 1 | + 1.71·2-s − 3-s + 0.939·4-s − 1.71·6-s + 0.654·7-s − 1.81·8-s + 9-s + 0.163·11-s − 0.939·12-s − 2.65·13-s + 1.12·14-s − 4.99·16-s + 3.86·17-s + 1.71·18-s − 3.47·19-s − 0.654·21-s + 0.280·22-s − 7.69·23-s + 1.81·24-s − 4.55·26-s − 27-s + 0.614·28-s + 29-s + 5.05·31-s − 4.93·32-s − 0.163·33-s + 6.62·34-s + ⋯ |
| L(s) = 1 | + 1.21·2-s − 0.577·3-s + 0.469·4-s − 0.699·6-s + 0.247·7-s − 0.642·8-s + 0.333·9-s + 0.0493·11-s − 0.271·12-s − 0.736·13-s + 0.299·14-s − 1.24·16-s + 0.936·17-s + 0.404·18-s − 0.798·19-s − 0.142·21-s + 0.0597·22-s − 1.60·23-s + 0.371·24-s − 0.892·26-s − 0.192·27-s + 0.116·28-s + 0.185·29-s + 0.907·31-s − 0.871·32-s − 0.0284·33-s + 1.13·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.71T + 2T^{2} \) |
| 7 | \( 1 - 0.654T + 7T^{2} \) |
| 11 | \( 1 - 0.163T + 11T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 - 3.86T + 17T^{2} \) |
| 19 | \( 1 + 3.47T + 19T^{2} \) |
| 23 | \( 1 + 7.69T + 23T^{2} \) |
| 31 | \( 1 - 5.05T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 6.17T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 4.04T + 53T^{2} \) |
| 59 | \( 1 + 0.328T + 59T^{2} \) |
| 61 | \( 1 + 5.72T + 61T^{2} \) |
| 67 | \( 1 + 3.51T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 1.12T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 7.89T + 83T^{2} \) |
| 89 | \( 1 - 5.04T + 89T^{2} \) |
| 97 | \( 1 + 8.49T + 97T^{2} \) |
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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
−8.565146581976008833915977555895, −7.81000755255237268293128194403, −6.76681579357223664876921348553, −6.12843890092574055077284946784, −5.36036177362494260385261017157, −4.70530082437166756509549457811, −3.96712978625600074749006692804, −3.00766383161831255895532087816, −1.81986020046948973544620214740, 0,
1.81986020046948973544620214740, 3.00766383161831255895532087816, 3.96712978625600074749006692804, 4.70530082437166756509549457811, 5.36036177362494260385261017157, 6.12843890092574055077284946784, 6.76681579357223664876921348553, 7.81000755255237268293128194403, 8.565146581976008833915977555895
